Some Problems Involving the p(x)-Polyharmonic Kirchhoff Operator

` DEGLI STUDI DI BARI “ALDO MORO” UNIVERSITA Dottorato di Ricerca in Matematica XXIV Ciclo – A.A. 2010/2011 Settore Scientifico-Disciplinare: MAT/05 – Analisi Matematica

Tesi di Dottorato

Some Problems Involving the p(x)-Polyharmonic Kirchhoff Operator Candidata:

Francesca COLASUONNO Supervisore della tesi:

Prof.ssa Patrizia PUCCI Tutore:

Dr. Lorenzo D’AMBROSIO Coordinatore del Dottorato di Ricerca:

Prof. Luciano LOPEZ

Dedicated to my parents, for giving me the greatest gift: my sisters.

Contents Introduction 1 Function Spaces 1.1 Variable Exponent Lebesgue and Sobolev Spaces . . . . . L,p(·) 1.2 An Equivalent Norm in W0 (Ω) . . . . . . . . . . . . . 1.3 The Vectorial Setting . . . . . . . . . . . . . . . . . . . . I

Existence for Elliptic Kirchhoff Systems

vii 1 1 6 9 13

2 Multiplicity of Solutions for p(x)-Polyharmonic Kirchhoff Systems 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries and Auxiliary Results . . . . . . . . . . . . 2.3 An Unbounded Sequence of Weak Solutions . . . . . . .

15 15 18 23

3 Multiplicity of Solutions for Eigenvalue p-Polyharmonic Kirchhoff Systems 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries and Auxiliary Results . . . . . . . . . . . . 3.3 Existence of Three Solutions . . . . . . . . . . . . . . . . 3.4 Complements to Chapter 3 . . . . . . . . . . . . . . . . .

29 29 31 34 52

II

79

Damped Evolution Kirchhoff Systems

4 Global Non-continuation and Qualitative Analysis for p(x)Polyharmonic Kirchhoff Systems 81 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 81

vi

Index

4.2 4.3 4.4 4.5 4.6

Preliminaries . . . . . . . . . . . . . Global Non-continuation Results . . Special Nonlinear External Damping Energy Estimates . . . . . . . . . . . Lifespan Estimates . . . . . . . . . .

. . . . .

. . . . .

5 Two Kirchhoff-Love Models 5.1 Introduction . . . . . . . . . . . . . . . . 5.2 Preliminaries and Energy Estimates . . . 5.3 Lifespan Estimates for the First Model . 5.4 Blow up at Infinity for the Second Model

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . .

86 90 98 100 113

. . . .

121 121 126 136 143

Introduction The aim of the thesis is to present new results on the existence, noncontinuation and qualitative aspects of solutions of higher order Kirchhoff systems. The whole work is based on the published articles [9, 10, 32, 33], which in some cases have been further extended in the framework of the dissertation. The original one-dimensional Kirchhoff equation was introduced by Kirchhoff himself [65] in 1883, to describe the oscillations of an elastic clamped string. This equation takes into account the effects of the changes in the length of the string, produced by transversal vibrations. In one dimension, a Kirchhoff type equation is of the form Z Eh L 2 %hutt − p0 + |ux | dx uxx + δut + f (x, u) = 0, (i) 2L 0 for t ≥ 0 and 0 < x < L, where u = u(t, x) is the lateral displacement at the time t and the space coordinate x, E the Young modulus, % the mass density, h the cross section area, L the length, p0 the initial axial tension, δ the resistance modulus and f an external force. The first equation introduced by Kirchhoff had δ = f = 0. It is worth mentioning that (i) received much attention after the work of J.L. Lions [77], in which a functional analysis framework was proposed for the problem. The Kirchhoff equation appears as a nonlinear extension of the classical d’Alembert wave equation, where the nonlinearity is the so-called Kirchhoff function, which in the case of (i) is given by Z L Z Eh L 2 M |ux | dx = p0 + |ux |2 dx. 2L 0 0

viii

Introduction

If p0 is equal to 0, the equation is called of degenerate type, otherwise it is of non-degenerate type. The nonlinearity M is an approximation, by averaging, of the classical von Kármán model, see [26] and references therein for more details. Due to the presence of this term, equation (i) is no longer a pointwise identity but it turns out to be a nonlocal equation. In two dimensions Kirchhoff equations model the oscillations of thin plates and the most usual plate operator is the biharmonic operator, see [17]. More generally, in [9, 10], we study Kirchhoff problems in higher dimension n ≥ 1 governed by the polyharmonic operator. In recent years there has been an increased interest in differential problems governed by higher order operators, like the polyharmonic operator, or by anisotropic operators, like the p(x)-Laplacian. It is impossible to cite all the papers involved in this branch of research, but we can not avoid mentioning the recent book [53] by Gazzola, Grunau and Sweers, the monograph [46] by Diening, Harjulehto, Hästö and R˚ uˇziˇcka and the survey paper [61], which testify to the special attention to the problems with variable exponents and to the corresponding generalized Lebesgue and Sobolev spaces. Furthermore, for the first time in literature Kratochvl and Nec˘as in [68] ˆ and then Drábek and Otani in [48], studied some problems involving the fourth order operator ∆2p u = ∆(|∆u|p−2 ∆u), called p-biharmonic operator. Following this trend and inspired by these papers, in [32] we introduced the 2L-order p(x)-operator ( ∆j |∆j u|p(x)−2 ∆j u , L = 2j, L j ∆p(x) u = j = 0, 1, 2, . . . j p(x)−2 j −div ∆ |D∆ u| D∆ u , L = 2j + 1, We point out that, with this definition, which is new also in the case p ≡ Const., we include a large class of more familiar differential operators, such as the p(x)-Laplacian −∆p(x) for L = 1, the polyharmonic operator (−∆)L for p ≡ 2, but also, for L = 2 and p ≡ Const. the p-biharmonic operator ∆2p . In what follows, we refer to ∆Lp(x) as the p(x)-polyharmonic operator. Recently we discovered that at the same time, also Lubyshev in [79] defined the operator ∆Lp u = ∆j (|∆j u|p−2 ∆j u), with p > 1, j = 1, 2, . . .

Introduction

ix

and L = 2j, that is only for L even. He proved the existence of multiple solutions of a nonlinear partial differential equation governed by the operator ∆Lp , with L even, under Dirichlet boundary conditions. In [32] as well as in Chapter 2 of the thesis, we study the existence of an unbounded sequence of weak solutions, for all L = 1, 2, . . . and p = p(x). For the p(x)-operators the natural setting is described by the variable exponent Sobolev spaces W L,p(·) (Ω). These spaces have been used in the last two decades to model various phenomena concerning nonhomogeneous materials, such as electrorheological fluids. Indeed, the behaviour of these anisotropic fluids cannot be modeled with sufficient accuracy using the classical Sobolev spaces, because of their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field. Usually in these problems, a variable exponent appears also in the growth condition of the nonlinearity. A field of application of problems with non-standard growth conditions is image restoring, where the variable nonlinearity is used to outline the borders of the true image and to eliminate the possible noise. Recently, the Helsinki group, a team of researchers studying variable exponent Lebesgue and Sobolev spaces, appeared in Finland; we refer to their web page http://www.helsinki.fi/∼pharjule/varsob/index.shtml for further details and a wide list of downloadable papers. In Chapter 1 we introduce the basic definitions and properties in the framework of the generalized Lebesgue and Sobolev spaces Lp(·) (Ω), W L,p(·) (Ω). For this introductory part we refer to [45, 46, 47, 49, 67]. With the exception of the problems studied in the Complements to Chapter 3, all our problems are higher order systems considered under Dirichlet boundary conditions and set in bounded domains Ω of Rn . L,p(·) (Ω)]d , where Hence, the natural solution space turns out to be [W0 more orders of differentiation are involved and several different equivalent norms are available. In the scalar setting d = 1 we prove, in L,p(·) Section 1.2, the equivalence of the standard norm of W0 (Ω) with the norm kDL ·kp(·) , by a Poincaré and a Caldéron-Zygmund type inequalities. The operator DL is given by ( ∆j u, if L = 2j, DL u = j = 0, 1, 2, . . . j D∆ u, if L = 2j + 1,

x

Introduction

and k · kp(·) denotes the Luxemburg norm of Lp(·) (Ω) if L is even and of [Lp(·) (Ω)]n if L is odd. Indeed, DL is a scalar or a vectorial operator, depending on whether L is even or odd. In order to have a Caldéron-Zygmund type inequality in these variable exponent Sobolev spaces, we need to require the domain Ω to have Lipschitz boundary. Therefore, in the whole thesis, the bounded domain Ω of Rn , n ≥ 1, has Lipschitz boundary in the problems set in the variable exponent Sobolev spaces, while this further assumption on Ω is not needed in the problems set in the classical Sobolev spaces. The only exception is the Robin problem studied in Section 3.4, in which the domain Ω is assumed to have C 1 boundary and n ≥ 2. In Part I we study the existence and multiplicity of solutions of higher order elliptic systems of Kirchhoff type. More precisely, in Chapter 2, which is based on our paper [32], we analyze the existence and multiplicity of solutions of the following problem ( M (IL (u)) ∆Lp(x) u = f (x, u) in Ω, (ii) α D u(x) = 0 for all α, with |α| ≤ L − 1, ∂Ω

where L = 1, 2, . . . , n ≥ 1 and p ∈ C(Ω) verifies the natural structural assumptions usually required in the generalized Sobolev spaces. Moreover u = (u1 , . . . , ud ) = u(x) represents the vectorial position and d ≥ 1. The L,p(·) functional IL : [W0 (Ω)]d → R+ 0 , known as p(x)-Dirichlet functional, is defined by Z |DL u|p(x) IL (u) = dx. p(x) Ω We remark that, since we are in the vectorial setting, ( Rd , if L = 2j, DL u = (DL u1 , . . . , DL ud ) ∈ Rnd , if L = 2j + 1, where, in the case L = 2j + 1, the d × n matrix DL u is seen as the ndvector obtained by putting the d lines of the Jacobian matrix of ∆j u one after the other. Furthermore, | · | denotes the d-Euclidean norm, if L is even, while it stands for the nd-Euclidean norm if L is odd. Analogously,

xi

Introduction

the p(x)-polyharmonic operator ∆Lp(x) u has to be understood as the dvectorial operator ∆j (|∆j u|p(x)−2 ∆j u1 ), . . . , ∆j (|∆j u|p(x)−2 ∆j ud ) , if L = 2j, j j j p(x)−2 j j p(x)−2 j D∆ ud , D∆ u1 , . . . ,−div ∆ |D∆ u| − div ∆ |D∆ u| if L = 2j + 1. In the linear case p ≡ 2, the vectorial p(x)-polyharmonic operator becomes the polyharmonic operator applied component by component, namely (−∆)L u = (−∆)L u1 , . . . , (−∆)L ud . We point out that the reason for studying the vectorial case is that the most interesting physical situation arises when d = 3 and u represents the position in the three dimensional space. The nonlinearity f : Ω × Rd → Rd in (ii) is a Carathéodory function with subcritical growth. Here, the variable exponent is not only related to the differential operator, but also to the growth of f . Indeed, among the other natural assumptions, we assume that |f (x, s)| ≤ C(1 + |s|q(x)−1 ) for all x ∈ Ω and s ∈ R, where C > 0 is a constant, q ∈ C(Ω) is a subcritical function, in the sense that q(x) ≤ p∗L (x) for all x ∈ Ω and p∗L is the variable critical exponent L,p(·) (Ω)]d ,→ [Lq(·) (Ω)]d , see Chapter 1 for further for the embedding [W0 details. Moreover, we require f = f (x, s) to be odd in s. This condition is useful to have a multiplicity result, without this symmetry assumption we would have just an existence result and we would have no information about the multiplicity of solutions. + The Kirchhoff function M : R+ 0 → R0 is allowed to be zero at zero, this means that we cover the degenerate case in our analysis. A prototype for M is given by the polynomial function M (τ ) = a + bγτ γ−1 ,

a, b ≥ 0,

a + b > 0,

γ ≥ 1.

(iii)

In standard literature the function M is supposed to be greater or equal to a positive constant, for instance in [4], Alves et al. prove the existence of positive solutions for a second order non-degenerate elliptic

xii

Introduction

Kirchhoff equation. The same result has been proved by Ma in [82] for a non-degenerate stationary Kirchhoff equation of the fourth order in one dimension. Only few attempts have been made to solve the degenerate case. In this direction we can mention Correa and Menezes [36], who give an existence result via Galerkin method for a possibly degenerate second order Kirchhoff problem. Problem (ii) has been already studied for the equations, that is when d = 1, in Section 4 of [32]. It has a variational structure which allows us to apply the symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [5], in order to prove the existence of an unbounded sequence L,p(·) of weak solutions in the space [W0 (Ω)]d . Chapter 3 contains and extends the results of our paper [33] to higher order systems. The new part in this thesis consists in the investigation, in Section 3.3, of the following eigenvalue system ( M (kukp )∆Lp u = λ{w(x)|u|p−2 u + f (x, u)} in Ω, (iv) Dα u(x) ∂Ω = 0 for all α, with |α| ≤ L − 1, where here p is constant, u = (u1 , . . . , ud ) = u(x), d ≥ 1, w is a positive weight with a suitable integrability property and the nonlinear perturbation f : Ω × Rd → Rd is now a (p − 1)-sublinear Carathéodory function. The vectorial p-polyharmonic operator ∆Lp coincides with the vectorial ∆Lp(x) , when p ≡ Const. The Kirchhoff function M is assumed to be of the special type given in (iii), with a > 0, so that in this case we consider only the non-degenerate case. For problem (iv) we determine the intervals of λ’s for which only the trivial solution exists and for which there are at least two nontrivial solutions. These intervals of λ’s are given in terms of the first eigenvalue λ1 of the main operator ∆Lp , which we prove to be positive as it is required in the proofs of the main theorems. This is the reason that leads us to study the eigenvalue problem in the classical Sobolev spaces rather than in the framework of the Sobolev spaces with variable exponents. Indeed, in this more general setting may arise that the first eigenvalue λ1 of the p(x)-polyharmonic operator is zero. For L = 1, in the work [51], Fan et al. prove that the first eigenvalue of the p(x)-Laplacian is positive if the function p satisfies some restrictive “monotonicity” properties obviously verified in the case p ≡ Const. However, as far as we know, no sufficient

xiii

Introduction

conditions have been given in literature for higher order p(x)-operators. In any case, under the additional hypothesis that λ1 is positive, it is possible to extend all the results also for the p(x)-polyharmonic operator. The techniques used here are based on a result by Arcoya and Carmona [7] which is an extension of the well-known three critical points theorem due to Pucci and Serrin [93, 94]; see also the recent book by Kristály, R˘adulescu and Varga [71] for further comments and applications. The results reported in the Complements to Chapter 3 are contained in the paper [33]. We analyze the following eigenvalue equation −divA(x, Du) = λ{w(x)|u|p−2 u + f (x, u)} in Ω, under Dirichlet boundary conditions u = 0 on ∂Ω, or under Robin boundary conditions A(x, Du) · ν + w1 (x)|u|p−2 u = λµg(x, u) on ∂Ω. Also in these problems p is constant. On f and w we require almost the same assumptions of problem (iv) with d = 1. Under Robin boundary conditions, we consider for simplicity only the case 1 < p < n, n ≥ 2. Moreover, w1 is a positive weight on the boundary ∂Ω and g : ∂Ω × R → R is a Carathéodory (p − 1)-sublinear function. A : Ω × Rn → R, A = A (x, ξ) is the potential of A with respect to ξ, it is strictly convex in ξ and such that c|ξ|p ≤ pA (x, ξ) ≤ C|ξ|p , for two suitable constants 0 < c ≤ C. All the assumptions are precisely stated in Chapter 3 and we refer to Section 3.4 for further details. The novelty in Section 3.4 is to cover the case in which the divergence operator is the usual p-Laplace operator for all p > 1, while usually in literature ∆p can be considered only when p ≥ 2, cf. [43]. This is possible thanks to the fact that we do not require A (x, ·) to be puniformly convex, but only strictly convex; see Remark 3.4.1. Also for these divergence type equations we determine the intervals of λ’s for which only the trivial solution exists and for which there are at least two nontrivial solutions, by using a modified version of two existence theorems given in [7].

xiv

Introduction

Part II of the thesis is devoted to evolution problems and contains non-continuation theorems and a qualitative analysis for the solutions of damped Kirchhoff systems involving time dependent source forces. In Chapter 4 we generalize some results obtained in Sections 3–6 of our paper [9] for a Kirchhoff system governed by the polyharmonic operator, to a class of problems involving the p(x)-polyharmonic operator. The system we analyze in R+ 0 × Ω is  utt + M (IL (u)) ∆Lp(x) u + N (IL−1 (u)) ∆L−1  p(x) u   p(x)−2 + µ|u| u + Q(t, x, u, ut ) = f (t, x, u), (v)    Dα u(t, x) =0 for all α, with |α| ≤ L − 1, + R0 ×∂Ω

where the function u = (u1 , . . . , ud ) = u(t, x) is the vectorial displacement, d ≥ 1 and µ is a non-negative parameter. The nonlinear damping Q is continuous in all its arguments and verifies (Q(t, x, u, v), v) ≥ 0 for all t, x, u, v, (vi) while f is an external source force, which is assumed to be continuous and derivable from a potential, f (x, s) = Ds F (x, s). With the term Q we model all the passive viscous type suppressions of the vibrations of the elastic structure. Therefore, Q has a stabilizing role, since it adsorbs the vibration energy. In contrast, f works against the continuation of solutions and makes easier the blow up. The term µ|u|p(x)−2 u represents a small perturbation. The nonlinearities M , N ∈ L1loc (R+ 0 ) are two non-negative possibly degenerate Kirchhoff functions, satisfying the condition γM (τ ) ≥ τ M (τ ),

ηN (τ ) ≥ τ N (τ ),

for all τ ∈ R+ 0

(vii)

with 1 ≤ η ≤ γ, where M and N are the primitives of M and N which vanish at zero. In the main Theorem 4.3.1 for problem (v), we show that if the initial energy is bounded above by a critical value, then there are no solutions of (v) which are global in time. The proof relies on concavity arguments combined with a new version of the potential well method, following a pioneering idea of Pucci and Serrin contained in [97], see also [15] for p(x)-Kirchhoff systems, where N = 0 and Q is less general. The interest in these kind of problems is even older. We can mention

xv

Introduction

the papers [16, 56, 72, 73, 74] in which existence, non-existence, stability and blow up results for nonlinear evolution equations are proved. Furthermore, for special functions M , Q and f we are able to perform energy estimates and a deep qualitative analysis of the solutions of (v), which lead us to establish an explicit a priori upper bound T0 for the lifespan T of maximal solutions. We remark here that the proofs of global non-continuation do not, in general, imply finite time blow up of solutions, since it is possible for the solutions to leave the domain of one of the differential operators before becoming unbounded. We refer to Section 4.1 for further details and comments on this topic. As a consequence of the non-existence theorem, we can prove in Corollary 4.6.3, that the maximal solutions of (v) blow up at finite time only under the further assumption that the lifespan T of the solutions coincides with the value T0 of the upper bound estimate. Finally, Chapter 5 is based on our papers [9, 10]. In this chapter we study two strongly damped polyharmonic Kirchhoff systems under homogeneous Dirichlet boundary conditions. For L = 1, 2, . . . the first problem we analyze in R+ 0 × Ω is utt +M kDL u(t, ·)k22 (−∆)L u+ N kDL−1 u(t, ·)k22 (−∆)L−1 u (viii) + µu + %(t)(−∆)L ut +Q(t)ut = f (t, x, u). The structural hypotheses on M and N are the same required in model (v). Here the external damping is linear in v, as it is standard in the KirchhoffLove plate models, so that with abuse of notation we write Q(t, x, u, v) = Q(t)v and we require that Q ∈ C 1 (R+ 0 ),

Q, −Q0 ≥ 0 in R+ 0,

which is clearly verified when Q does not depend on t, as it usually happens in literature, see e.g. [2, 57, 59, 90, 107, 108, 109]. Moreover, the term %(t)(−∆)L ut represents the internal material damping of KelvinVoigt type of the body structure, which is always present, even if small, in real materials as long as the system vibrates. Also on % we suppose the natural restriction % ∈ C 1 (R+ 0 ),

%, −%0 ≥ 0 in R+ 0,

which is automatic in the standard case in which %(t) ≡ %0 > 0 in R+ 0. We note in passing that, if % ≡ 0, problem (viii) reduces to a special case

xvi

Introduction

of (v) when p ≡ 2. The presence of the internal damping term makes the analysis more delicate. This is the main reason why we replace the general operator ∆Lp(x) with the linear operator (−∆)L to handle this case. Indeed, in this setting, it is possible to apply the concavity method in order to show that (viii) has no global solutions. With this technique we provide, in Theorem 5.3.1, an explicit upper bound for the lifespan of maximal solutions. We use a different approach to study the following model in R+ 0 ×Ω utt +M kDL u(t, ·)k22 (−∆)L u + N kDL−1 u(t, ·)k22 (−∆)L−1 u (ix) + µu + %(t)K kDL u(t, ·)k22 (−∆)L ut +Q(t, x, u, ut ) = f (t, x, u), under homogeneous Dirichlet boundary conditions. The main Kirchhoff function M is of the special form (iii), while N satisfies the same condition required for problem (v). Both M and N are possibly degenerate. The third Kirchhoff function K ∈ L1loc (R+ 0 ) is definitively strictly positive and is related to M by the assumption that for every σ > 0 there is cσ > 0 such that M (τ ) ≥ cσ K (τ ) for all τ ≥ σ, where K is the primitive of K which vanishes at zero. On Q we require (vi) and on % we assume % ∈ C 1 (R+ 0 ),

% 6≡ 0 and %, %0 ≥ 0 in R+ 0,

which is clearly verified in the standard case % ≡ %0 > 0. Problem (ix) reduces to a special case of model (v) when p ≡ 2 and %(t)K(kDL u(t, ·)k22 ) ≡ 0 in R+ 0 . As already said, for problem (v) we prove, under suitable hypotheses, that no global solutions exist. This is possible thanks to a certain interaction between the external force and the damping term. A fairly natural question is to understand what happens if the additional damping %(t)K(kDL u(t, ·)k22 ) (−∆)L ut is included in the system. In literature this problem has been investigated, for instance, in [40] by D’Ancona and Shibata who study the global existence of analytic solutions of problems describing nonlinear viscoelastic materials with short memory. The models studied in [40] are a special case of (ix) when L = 1, N ≡ 0, % ≡ 1, Q ≡ 0 and f ≡ 0. Furthermore, in [87] Ono establishes the existence of a global solution for a subcase of (ix), when Q ≡ 0, % ≡ 1 and K ≡ 1, assuming Eu(0) limited above, see also [89, 90].

Introduction

xvii

Indeed, a strong action of dissipative terms could make easier the existence of global solutions, since they play the role of stabilizing terms and their smoothing effect makes more difficult the blow up. For all these reasons, in Theorem 5.4.2, we study the behaviour of the solutions of (ix) at infinity and we show that if the initial energy is bounded above by a critical value, then every global solution blows up when the time goes to infinity. The Structure of the Thesis The work produced in this thesis is based on the papers [9, 10, 32, 33], which have been extended in several directions. The thesis is divided into two parts and into five chapters. All the problems studied in this thesis are set in bounded domains Ω of Rn . We resume here the structure of the thesis. Chapter 1 is devoted to the Lebesgue and Sobolev spaces with variable exponents. In this chapter we report the basic results for this setting and we conclude by giving a new proof of the equivalence between the L,p(·) standard norm of W0 (Ω) and the norm which will be used throughout the dissertation. Moreover, we present similar results for the vectorial L,p(·) (Ω)]d , d ≥ 1. setting [W0 Part I contains Chapters 2 and 3 and is devoted to the existence and multiplicity of solutions of higher order elliptic problems involving the Kirchhoff function. In Chapter 2 we present some new results based on [32]. We introduce the definition of the p(x)-polyharmonic operator and we prove the existence of an unbounded sequence of weak solutions of an elliptic p(x)-Kirchhoff system under Dirichlet boundary conditions, by using the symmetric mountain pass theorem by Ambrosetti and Rabinowitz. In Chapter 3 we extend the work [33] to some eigenvalue p-Kirchhoff problems. We determine the intervals of λ’s for which there exist nontrivial solutions. The main tool is a new version by Arcoya and Carmona of the three critical points theorem by Pucci and Serrin. The complements to this chapter contain the results of [33]. We analyze an eigenvalue equation governed by a divergence type operator, under Dirichlet boundary conditions or under Robin boundary conditions. Part II is divided into Chapter 4 and Chapter 5. In this part we study some damped and strongly damped evolution systems of Kirchhoff type,

xviii

Introduction

governed by time dependent source forces, under homogeneous Dirichlet boundary conditions. Chapter 4 takes inspiration from [9], more precisely we generalize the polyharmonic problem studied in Sections 3–6 of that paper, to a Kirchhoff system involving the p(x)-polyharmonic operator. This forces us to change the functional setting and to use the variable exponent Sobolev spaces. For this type of problem we prove that do not exist any solutions which are global in time. In some special cases we are also able to give an explicit a priori upper bound for the lifespan of maximal solutions. Finally, in Chapter 5 we report some of our results already appeared in the articles [9, 10], concerning two different strongly damped KirchhoffLove problems under homogeneous Dirichlet boundary conditions. These models are both governed by the polyharmonic operator and contain a term which takes into account the internal damping of a vibrating system. For the first problem we study the non-continuation of the solutions via a concavity argument, while for the second problem we prove that all global solutions blow up at infinity. The unpublished results of Chapters 3 and 4 provide the subject for two forthcoming papers. Acknowledgements I am deeply indebted to my scientific supervisor Prof. Patrizia Pucci not only for her inestimable mathematical teachings, but also for her extraordinary helpfulness, enthusiasm and all the time she devoted to me. Working with her has been one of the most enriching experiences of my life. I would like to express my greatest gratitude to her. Special thanks go to Dr. Giuseppina Autuori for her mathematical suggestions and her practical and psychological support during the period I spent in Perugia. Her friendship has made the city of Perugia a second home for me. I also wish to thank my tutor Dr. Lorenzo D’Ambrosio for his precious advice and the important contacts he provided for my mathematical development. Last but not least, my gratitude goes also to Prof. Csaba Varga, for the fruitful discussions, in particular about Chapter 3 of this thesis, which would have been much poorer without his suggestions.

Chapter 1

Function Spaces In this chapter we introduce the definitions and basic properties of variable exponent Lebesgue and Sobolev spaces. We refer essentially to the monograph [46] by Diening et al., as well as to the pioneering papers [45, 47, 49, 67]. The classical Lebesgue and Sobolev spaces will be obtained as particular cases of the more general function spaces presented below.

1.1

Variable Exponent Lebesgue and Sobolev Spaces

Let Ω ⊂ Rn be a bounded domain. Define for all h ∈ C(Ω) h+ = max h(x)

and

h− = min h(x) x∈Ω

x∈Ω

and put C+ (Ω) = {h ∈ C(Ω) : h− > 1}. From now on in this section p ∈ C+ (Ω) is a fixed variable exponent. The variable exponent Lebesgue space Lp(·) (Ω) is defined by Z p(x) p(·) |u(x)| dx < ∞ L (Ω) = u : Ω → R : Ω

and is endowed with the so-called Luxemburg norm ( ) Z u(x) p(x) kukp(·) = inf λ > 0 : dx ≤ 1 . λ Ω

(1.1.1)

2

Chapter 1. Function Spaces

Note that, when p ≡ Const., the Luxemburg norm k · kp(·) coincide with the standard norm k · kp of the Lebesgue space Lp (Ω). Since here 0 < |Ω| < ∞, the following result holds. Theorem 1.1.1 (Theorem 2.8 of [67]). If h ∈ C(Ω) and 1 ≤ h ≤ p in Ω, then the embedding Lp(·) (Ω) ,→ Lh(·) (Ω) is continuous and the norm of the embedding operator does not exceed |Ω| + 1. Let p0 be the function obtained by conjugating the exponent p pointwise, that is 1/p(x) + 1/p0 (x) = 1 for all x ∈ Ω, then p0 belongs to C+ (Ω). Theorem 1.1.2 (Theorem 2.5 and Corollaries 2.7 and 2.12 of [67]). 0 Lp(·) (Ω) is a separable, reflexive, Banach space and Lp (·) (Ω) is its dual space. Theorem 1.1.3 (Theorem 2.1 of [67]). For any u ∈ Lp(·) (Ω) and v ∈ 0 Lp (·) (Ω), the following Hölder type inequality is valid Z |u(x)v(x)| dx ≤ cp(·) kukp(·) kvkp0 (·) , (1.1.2) Ω

with cp(·) = 1 + 1/p− − 1/p+ . Let h be a function in C(Ω). An important role in manipulating the generalized Lebesgue and Sobolev spaces is played by the h(·)-modular of Lh(·) (Ω), which is the convex function ρh(·) : Lh(·) (Ω) → R defined by Z |u(x)|h(x) dx. (1.1.3) ρh(·) (u) = Ω

Theorem 1.1.4 (Theorems 1.3 and 1.4 of [52]). If u, (uk )k are in Lh(·) (Ω), with 1 ≤ h− ≤ h+ < ∞, then the following relations hold: kukh(·) < 1 (= 1; > 1) ⇔ ρh(·) (u) < 1 (= 1; > 1), kukh(·) ≥ 1 kukh(·) ≤ 1

h

h

⇒

− + kukh(·) ≤ ρh(·) (u) ≤ kukh(·) ,

⇒

h+ kukh(·)

≤ ρh(·) (u) ≤

(1.1.4)

h− kukh(·) ,

and kuk − ukh(·) → 0 ⇔ ρh(·) (uk − u) → 0 ⇔ uk → u in measure in Ω and ρh(·) (uk ) → ρh(·) (u). In particular, ρh(·) is continuous in Lh(·) (Ω), and if furthermore h ∈ C+ (Ω), then ρh(·) is weakly lower semicontinuous.

3

Lemma 1.1.5 (Lemma 2.1 of [49]). Let h ∈ L∞ (Ω) be such that 1 ≤ (hp)(x) ≤ ∞ for a.a. x ∈ Ω. Let u ∈ Lhp(·) (Ω), u 6≡ 0. Then h

h

h

h

+ − , if kukhp(·) ≥ 1, ≤ k |u|h(·) kp(·) ≤ kukhp(·) kukhp(·) + − kukhp(·) ≤ k |u|h(·) kp(·) ≤ kukhp(·) , if kukhp(·) ≤ 1.

In particular, if h ≡ Const., then k |u|h khp(·) = kukhhp(·) . n PnBy α = (α1 , ..., αn ) ∈ N0 we denote a multi-index, with length |α| = i=1 αi and corresponding partial differentiation

Dα =

∂ |α| . ∂xα1 1 ...∂xαnn

For L = 1, 2, . . . we introduce the variable exponent Sobolev space W L,p(·) (Ω), defined by W L,p(·) (Ω) = u ∈ Lp(·) (Ω) : Dα u ∈ Lp(·) (Ω) for all α ∈ Nn0 , with |α| ≤ L and endowed with the standard norm X kukW L,p(·) (Ω) = kDα ukp(·) . |α|≤L

From now on in the chapter we also assume that p ∈ C+log (Ω), where C+log (Ω) = h ∈ C+ (Ω) : h is logarithmic Hölder continuous and h is logarithmic Hölder continuous, if there exists K > 0 such that for all x, y ∈ Ω, with 0 < |x − y| ≤ 1/2, |p(x) − p(y)| ≤ − L,p(·)

K . log |x − y|

(1.1.5)

The space W0 (Ω) denotes the closure in W L,p(·) (Ω) of the set of all W L,p(·) (Ω)-functions with compact support. As shown in [46, CorolL,p(·) lary 11.2.4], thanks to (1.1.5), the space W0 (Ω) coincides with the completion of C0∞ (Ω) with respect to the norm k · kW L,p(·) (Ω) .

4

Chapter 1. Function Spaces L,p(·)

Theorem 1.1.6 (Theorem 8.1.13 of [46]). W0 (Ω) is a separable, uniformly convex (and thus reflexive) Banach space. Also in this context it is possible to prove a Poincaré inequality. L,p(·)

Theorem 1.1.7. For every u ∈ W0

(Ω) the following inequality holds X ≤ CL kDα ukp(·) , (1.1.6)

kukW L,p(·) (Ω)

|α|=L

where CL = CL (p, n, Ω). Proof. Proceed by induction on L = 1, 2, . . . . By Theorem 4.1 of [60] it results that there exists a positive constant C = C(p, n, Ω) such that 1,p(·) for all u ∈ W0 (Ω)

n X

∂u

kukp(·) ≤ C

∂xi . p(·) i=1 P Hence, kukW 1,p(·) (Ω) ≤ C1 |α|=1 kDα ukp(·) , with C1 = C + 1. Suppose by L,p(·)

induction that (1.1.6) holds for L − 1, then for all u ∈ W0 X kukW L,p(·) (Ω) = kukW L−1,p(·) (Ω) + kDα ukp(·)

(Ω)

|α|=L

X

≤ CL−1

kDα ukp(·) +

|α|=L−1

X

≤ CL−1 C = CL

kDα ukp(·)

|α|=L α

kD ukp(·) +

|α|=L

X

X X

kDα ukp(·)

|α|=L

kDα ukp(·) ,

|α|=L

where CL = CL−1 C +1. Of course, in the last inequality, we have used the L,p(·) 1,p(·) basis of the induction, since u ∈ W0 (Ω) implies that Dα u ∈ W0 (Ω) if |α| = L − 1. This concludes the proof. 2 L,p(·)

Hence, an equivalent norm for the space W0 (Ω) is given by X kukW L,p(·) (Ω) = kDα ukp(·) . 0

|α|=L

5

Let p∗L denote the critical variable exponent related to p, defined for all x ∈ Ω by the pointwise relation

p∗L (x)

 

np(x) , if n > Lp(x), = n − Lp(x)  ∞, if 1 ≤ n ≤ Lp(x).

(1.1.7)

Theorem 1.1.8. Let h be of class C(Ω). If p− ≥ n/L and h ≥ 1 in Ω, then the embedding L,p(·)

W0

(Ω) ,→ Lh(·) (Ω)

is compact. If p+ < n/L and 1 ≤ h(x) ≤ p∗L (x) for all x ∈ Ω, then the embedding L,p(·)

W0

(Ω) ,→ Lh(·) (Ω)

is continuous, more precisely, there exists Sh+ = Sh+ (n, p, L, Ω) > 0 such that kukh(·) ≤ Sh+ kukW L,p(·) (Ω) 0

L,p(·)

for all u ∈ W0

(Ω).

(1.1.8)

Finally, if p+ < n/L and 1 ≤ h(x) < p∗L (x) for all x ∈ Ω, or equivalently 1 ≤ h− ≤ h+ < (p∗L )− , then the embedding L,p(·)

W0

(Ω) ,→ Lh(·) (Ω)

is compact. L,p(·)

L,p

Proof. Being p(x) ≥ p− in Ω, the embedding W0 (Ω) ,→ W0 − (Ω) is continuous, by Theorem 1.1.1. Moreover, since p− ≥ n/L and h+ < L,p ∞, the embedding W0 − (Ω) ,→ Lh+ (Ω) is compact and in turn also L,p(·) L,p W0 − (Ω) ,→ Lh(·) (Ω) is compact. Hence, W0 (Ω) ,→ Lh(·) (Ω) compactly, cf. [67, Theorem 3.8]. For a proof of (1.1.8) in the case L = 1, see [46, Theorem 8.3.1 and Corollary 8.3.2], then proceed by induction on L as in [1]. For the last part of the theorem, we refer to [45, Theorem 5.7], [52, Theorem 2.3] and [84, Proposition 3.3]. 2

6

1.2

Chapter 1. Function Spaces L,p(·)

An Equivalent Norm in W0

(Ω)

In what follows we require that Ω ⊂ Rn is a bounded domain with Lipschitz boundary and p ∈ C+log (Ω). Under this assumption, when L = 2, as a consequence of the main Caldéron-Zygmund result [47, Theorem 6.4], there exists a constant κ2 = κ2 (n, p) > 0 such that kukW 2,p(·) (Ω) ≤ κ2 k∆ukp(·) 0

2,p(·)

for all u ∈ W0

(Ω).

(1.2.1)

Remark 1.2.1. When p ≡ Const., inequality (1.2.1) holds without the further assumption that Ω has Lipschitz boundary, see [56, Corollary, 9.10]. Throughout the thesis, the operator DL is defined for every fixed L = 0, 1, 2, . . . by ( ∆j u, if L = 2j, DL u = j = 0, 1, 2, . . . , (1.2.2) j D∆ u, if L = 2j + 1, for all u ∈ W L,p(·) (Ω). Note that DL is a scalar operator if L is even and it is a vectorial operator of dimension n if L is odd. Hence, we need to define a norm also in the vectorial variable exponent Lebesgue space. We endow [Lp(·) (Ω)]n with the norm kvkp(·) =

n X

kvi kp(·) ,

(1.2.3)

i=1

where v = (v1 , . . . , vn ). With abuse of notation, where it is clear from the context, we use the same symbol k · kp(·) to denote both the standard Luxemburg norm in the scalar space Lp(·) (Ω) and the norm defined in (1.2.3) for the vectorial space [Lp(·) (Ω)]n . Proposition 1.2.2. For all L = 1, 2, . . . there exists a positive constant κL = κL (n, p) such that kukW L,p(·) (Ω) ≤ κL kDL ukp(·) 0

where DL is defined in (1.2.2).

L,p(·)

for all u ∈ W0

(Ω),

(1.2.4)

7

Proof. We distinguish two cases depending on whether L is even or odd. Case L = 2j. Proceed by induction on j = 1, 2, . . . . The case j = 1 is true by the consequence of the Caldéron-Zygmund inequality (1.2.1). Suppose that the inequality (1.2.4) holds for j ≥ 1 and show that it is true also for j + 1. Let u be in C0∞ (Ω) and L = 2(j + 1), then we get X X X kukW L,p(·) (Ω) = kDβ (Dα u)kp(·) ≤ κ2 kDα (∆u)kp(·) 0

|α|=2j |β|=2

|α|=2j

≤ κ2 κ2j kD2j (∆u)kp(·) = κL kDL ukp(·) , where κL = κ2 κ2j , the first inequality follows by (1.2.1) and the second by induction. We conclude the proof using density arguments. Indeed, L,p(·) (Ω) with respect to the norm k· by the fact that C0∞ (Ω) is dense in W0 L,p(·) kW L,p(·) (Ω) thanks to (1.1.5), for all u ∈ W0 (Ω) there exists a sequence ∞ (uk )k ⊂ C0 (Ω) such that limk→∞ kuk −ukW L,p(·) (Ω) = 0, thus in particular limk→∞ kuk kW L,p(·) (Ω) = kukW L,p(·) (Ω) and limk→∞ kDL uk kp(·) = kDL ukp(·) . 0 0 Case L = 2j + 1. When L = 1, then (1.2.4) is trivial. Proceeding by induction on j = 1, 2, . . . , we suppose that the inequality holds for j and we show that it is true also for j + 1, that is for L = 2j + 3. Indeed for all u ∈ C0∞ (Ω), by (1.2.1) X X X kukW L,p(·) (Ω) = kDβ (Dα u)kp(·) = κ2 kDα (∆u)kp(·) 0

|α|=2j+1 |β|=2

|α|=2j+1

≤ κ2 κ2j+1 kD2j+1 (∆u)kp(·) = κL kDL ukp(·) where κL = κ2 κ2j+1 . The proof is completed using again a density argument. 2 L,p(·)

Clearly, for all u ∈ W0

(Ω) the converse inequality

kDL ukp(·) ≤ kukW L,p(·) (Ω) , 0

holds. Therefore, as a consequence of Proposition 1.2.2, we endow the L,p(·) space W0 (Ω) with the equivalent norm k · k = kDL · kp(·) . We present here an important result in the special case p ≡ Const.

8


Proposition 1.2.3. If 1 < p < ∞, then the space (W0L,p (Ω), k · k) is uniformly convex. Proof. Fix ε ∈ (0, 2) and let u, v ∈ W0L,p (Ω) be such that kuk = kvk = 1 and ku − vk ≥ ε. First, consider the case p ∈ [2, ∞). By Lemma 2.27 of [1], we have that for all w, z ∈ C z + w p z − w p 1 p p + 2 2 ≤ 2 (|z| + |w| ). Hence,

p p

u + v p u − v p

+

= DL u + v + DL u − v

2

2

2 2 p p Z DL u + DL v p DL u − DL v p dx = + 2 2 Ω Z 1 ≤ (|DL u|p + |DL v|p ) dx 2 Ω 1 = (kukp + kvkp ) = 1. 2 This implies that

p

u + v p

≤1− ε

2 2

and the proof of this case is concluded. If p ∈ (1, 2), then by Theorem 2.7 of [1]

p0 p0

u + v u − v

+ DL

DL

2 2 p−1 p−1

p0 p0

u + v u − v

+ DL ≤ DL

2 2

p−1

Moreover, by Lemma 2.27 of [1] 0 0 1/(p−1) z + w p z − w p + ≤ 1 (|z|p + |w|p ) 2 2 2

(1.2.5) .

9

for all w, z ∈ C. Hence,

p0

u + v

DL +

2

p0 u − v DL

2

p−1

Z

1 ≤ (|DL u|p + |DL v|p ) dx Ω 2 1/(p−1) 1 1 p p , = kuk + kvk 2 2

1/(p−1)

(1.2.6)

0

being |DL φ|p ∈ Lp−1 (Ω) for all φ ∈ W0L,p (Ω). Note that, if φ ∈ W0L,p (Ω), 0 0 0 then k|DL φ|p kp−1 = kDL φkpp = kφkp , therefore, combining together (1.2.5) and (1.2.6), we get

0

0 1/(p−1)

u + v p u − v p

+

≤ 1 kukp + 1 kvkp

2

2 2 2 and so

0

0 p0

u + v p

u − v p

≤1−

≤1− ε ,

2

2 2 which concludes the proof.

1.3

2

The Vectorial Setting

In this section we prepare the notation and the setting for the vectorial L,p(·) spaces with variable exponents [W0 (Ω)]d , with d ≥ 1, which will be useful throughout the dissertation. Let d be an integer, d ≥ 1, p ∈ C+log (Ω) and Ω a bounded domain of Rn with Lipschitz boundary. As already done in the previous section, we endow the vectorial space [Lp(·) (Ω)]d with the norm kuk[Lp(·) (Ω)]d =

d X

kui kp(·) ,

i=1

for all u = (u1 , . . . , ud ) ∈ [Lp(·) (Ω)]d , where k · kp(·) is the Luxemburg norm of the scalar space Lp(·) (Ω), defined in (1.1.1).

10


Given a multi-index α = (α1 , ..., αn ) ∈ Nn0 , for each vector valued map u : Ω → Rd its |α|-order derivative is ∂ |α| u1 ∂ |α| ud α . D u= , . . . , α1 ∂xα1 1 ...∂xαnn ∂x1 ...∂xαnn For every L = 1, 2, . . . we equip the vectorial space [W L,p(·) (Ω)]d with the norm ! d d X X X kuk[W L,p(·) (Ω)]d = kui kW L,p(·) (Ω) = kDα ui kp(·) i=1

X

=

|α|≤L

i=1

kDα uk[Lp(·) (Ω)]d .

|α|≤L L,p(·)

The space [W0 (Ω)]d is the completion of [C0∞ (Ω)]d , with respect to the norm k · k[W L,p(·) (Ω)]d , being p ∈ C+log (Ω). Clearly, by Theorem 1.1.7, we have that also in the vectorial setting there exists a positive constant CL such that kuk[W L,p(·) (Ω)]d ≤ CL

d X X

kDα ui kp(·) = CL

i=1 |α|=L L,p(·)

for all u ∈ [W0 given by

X

kDα uk[Lp (·)(Ω)]d

|α|=L L,p(·)

(Ω)]d , therefore an equivalent norm in [W0

kuk[W L,p(·) (Ω)]d =

X

0

(Ω)]d is

kDα uk[Lp (·)(Ω)]d .

|α|=L

Moreover, by Proposition 1.2.2, it results that there exists a constant L,p(·) κL = κL (n, p) > 0 such that for every u ∈ [W0 (Ω)]d ( P if L = 2j, κL di=1 kDL ui kLp(·) (Ω) , Pd kuk[W L,p(·) (Ω)]d ≤ 0 κL i=1 kDL ui k[Lp(·) (Ω)]n , if L = 2j + 1 ( d, if L = 2j, = κL kDL uk[Lp (·)(Ω)]s , with s = nd, if L = 2j + 1. L,p(·)

Hence, throughout the thesis, we endow the space [W0 the equivalent norm k · k = kDL · k[Lp(·) (Ω)]s .

(Ω)]d with

11

L,p(·)

Remark 1.3.1. Clearly, the space ([W0 (Ω)]d , k · k) is a reflexive Banach space. Moreover, the embedding Theorem 1.1.8 proved for W L,p(·) (Ω) continue to hold also in the vectorial setting, since, in general, if X, Y are two Banach spaces such that X ⊂ Y and the embedding X ,→ Y is continuous (resp. compact), then also the embedding X d ,→ Y d is continuous (resp. compact).

Part I

Existence for Elliptic Kirchhoff Systems

Chapter 2

Multiplicity of Solutions for p(x)-Polyharmonic Kirchhoff Systems In this chapter we prove the existence of infinitely many solutions of a wide class of Dirichlet problems involving the p(x)-polyharmonic operator.

2.1

Introduction

In recent years, there has been an increasing interest in equations involving p(x)-Laplace operators, motivated by the modeling of thermo convective flows of non-Newtonian fluids, as well as the electrorheological fluids (sometimes referred to as smart fluids) characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field. The p(x)-Laplace operators possess more complicated nonlinearities than p-Laplace operators, mainly due to the fact that they are not homogeneous. Consider the problem ( M (IL (u)) ∆Lp(x) u = f (x, u) in Ω, (2.1.1) α D u(x) = 0 for all α, with |α| ≤ L − 1, ∂Ω

where L = 1, 2, . . . , Ω ⊂ Rn is a bounded domain with Lipschitz boundary, n ≥ 1, u = (u1 , . . . , ud ) = u(x) with d ≥ 1 is the vectorial position,

16

Chapter 2. p(x)-Polyharmonic Elliptic Kirchhoff Systems

α is a multi-index and p ∈ C+log (Ω) is such that either p+ < n/L or p− ≥ L,p(·) n/L. For L = 1, 2, . . . , the Dirichlet functional IL : [W0 (Ω)]d → R+ 0 is defined by Z |DL u|p(x) IL (u) = dx, (2.1.2) p(x) Ω where DL is the main L-order differential operator introduced in (1.2.2), applied component by component, namely DL u = (DL u1 , . . . , DL ud ) and | · | stands for the d-Euclidean norm, if L is even, while it denotes the nd-Euclidean norm if L is odd. L,p(·) For every φ ∈ W0 (Ω), the p(x)-polyharmonic operator ∆Lp(x) , for j = 0, 1, 2, . . . , is given by ( ∆j |∆j φ|p(x)−2 ∆j φ , if L = 2j, L j ∆p(x) φ = −div ∆ |D∆j φ|p(x)−2 D∆j φ , if L = 2j + 1. Since we are in the vectorial setting, for every u = (u1 , . . . , ud ) in the L,p(·) space [W0 (Ω)]d , ∆Lp(x) u has to be understood as the d-vector  j j p(x)−2 j j j p(x)−2 j  ∆ (|∆ u| ∆ u ), . . . , ∆ (|∆ u| ∆ u ) , if L = 2j,  1 d     (2.1.3) − div ∆j |D∆j u|p(x)−2 D∆j u1 , . . .       , if L = 2j + 1. . . . , −div ∆j |D∆j u|p(x)−2 D∆j ud In the linear case p ≡ 2, the vectorial p(x)-polyharmonic operator becomes the polyharmonic operator applied component by component, namely (−∆)L u = (−∆)L u1 , . . . , (−∆)L ud . Let us introduce the number pL = (p∗L )− /p+ where p∗L denotes the critical variable exponent related to p, defined in (1.1.7). The Kirchhoff + function M : R+ 0 → R0 verifies the structural assumption (M ) M is continuous and there exists a number γ ∈ [1, pL ), such that for all τ ∈ R+ 0 Z τ M (z)dz, γM (τ ) ≥ τ M (τ ), where M (τ ) = 0

17

and for all σ > 0 there exists κ = κ(σ) > 0 such that M (τ ) ≥ κ for all τ ≥ σ. We assume that f : Ω × Rd → Rd is a Carathéodory function (i.e. f (·, s) is measurable for every s ∈ Rd and f (x, ·) is continuous for a.a. x ∈ Ω), satisfying the following properties (f1 ) f is odd in s, i.e. f (x, −s) = −f (x, s) for all x ∈ Ω and s ∈ Rd ; (f2 ) there exist C > 0 and q ∈ C+ (Ω), with γp+ < q− ≤ q+ < (p∗L )− , such that |f (x, s)| ≤ C(1 + |s|q(x)−1 ) for all x ∈ Ω and s ∈ Rd ; (f3 ) there exist three constants µ > γp+ , c > 0 and U ≥ 0, such that c ≤ µF (x, s) ≤ s · f (x, s) for all x ∈ Ω and |s| ≥ U , where f (x, s) = Ds F (x, s), F (x, 0) = 0; (f4 ) lim |s|−γp+ F (x, s) = 0 uniformly in x ∈ Ω. |s|→0

Conditions (f1 )–(f4 ) are somehow related to [21] when M ≡ 1 and L = 1. We note that condition (f1 ) is useful to have a multiplicity result. Without this symmetry assumption we have just the existence result using the classical mountain pass theorem and we have no information about the multiplicity of solutions. Furthermore, in (f2 ) the growth of f (x, ·) is related to the variable exponent q. An interesting field of application for equations with variable exponent growth conditions is image restoring, where the variable nonlinearity is used to outline the borders of the true image and to eliminate the possible noise; details can be found in [6, 38]. A very special Kirchhoff function verifying (M ) is given by

a, b ≥ 0,

M (τ ) = a + bγτ γ−1 , ( ∈ (1, pL ), a + b > 0, γ = 1,

if b > 0, if b = 0.

(2.1.4)

When M is of the type (2.1.4), problem (2.1.1) is said to be non-degenerate when a > 0 and b ≥ 0, while it is called degenerate if a = 0 and b > 0.

18


When a > 0 and b = 0, problem (2.1.1) reduces to the usual well-known quasilinear elliptic system. The existence of positive solutions of nondegenerate Kirchhoff type problems is treated in [4, 35, 81] for L = 1 and p ≡ Const. The main novelty in this chapter is to cover the degenerate case of (2.1.1), that is the case in which M can be zero at zero, for the first time in literature. Fourth order problems with nonlinear boundary conditions involving third order derivatives has been considered by several authors (see [82] and the references therein), but in those cases the possibly degenerate Kirchhoff function M multiplies lower order terms rather than the leading fourth order term. Existence and multiplicity results for problem (2.1.1) when L = 1, d = 1 and M is strictly positive have already been given in [38] via variational methods, while in [39] the authors use a three critical point theorem due to Ricceri to solve the problem. Moreover, in [39] the authors assume p− > n, a condition which does not cover the natural case n = 3 and p ≡ 2. This situation has been considered in [25], but only for non-degenerate Kirchhoff equations. Problem (2.1.1), when p ≡ Const., models several interesting phenomena studied in mathematical physics, even in the case n = 1. Indeed, the original evolution equation was introduced by Kirchhoff in [65] to describe the nonlinear vibrations of an elastic string. Here we study a stationary version of Kirchhoff type problems, where u = u(x) is the lateral displacement at the space coordinate x and M is typically a line with positive slope. Our main result allows M to have this property. Further details and physical models described by the classical Kirchhoff theory can be found in [14, 103]. The results of this chapter are an extension to the case of the systems of Section 4 of [32], in which only the case d = 1 was treated.

2.2

Preliminaries and Auxiliary Results

In this section we collect a series of results and notations which will be used throughout the chapter. From here on in the section, we denote by X a real Banach space and by X ? its topological dual space. Definition 2.2.1. Let J : X → R be a functional of class C 1 (X). A sequence (uk )k ⊂ X is said to be a Palais-Smale sequence for J, (PS) sequence for shortness, if (J(uk ))k is bounded and J 0 (uk ) → 0 as k → ∞.

19

The functional J satisfies the Palais-Smale condition, (PS) condition, if every (PS) sequence admits a convergent subsequence. Definition 2.2.2. The functional J 0 : X → X ? verifies the (S+ ) property if for any sequence (uk )k ⊂ X which converges weakly to u in X and such that lim suphJ 0 (uk ), uk − ui ≤ 0, k→∞

it results that (uk )k converges strongly to u in X. Theorem 2.2.3 (Theorem 11.5 of [63]). Assume that X has infinite dimension and let J ∈ C 1 (X) be a functional satisfying the (PS) condition as well as the following three properties: (i) J(0) = 0 and there exist two constants r, α > 0 such that J ∂Br ≥ α; (ii) J is even; e ⊂ X there exists R = R(X) e > (iii) for all finite dimensional subspaces X 0 such that e J(u) ≤ 0 for all u ∈ X \ BR (X), e = {u ∈ X e : kuk < R}. where BR (X) Then J possesses an unbounded sequence of critical values characterized by a minimax argument. L,p(·)

As already said in Section 1.3, we endow the solution space [W0 (Ω)]d with the norm k · k = kDL · k[Lp(·) (Ω)]s , where s = d if L is even and s = nd if L is odd. For simplicity in notation we drop the exponent d, nd in all the functional spaces involved in the treatment and we denote L,p(·) L,p(·) by W0 (Ω) the vectorial space [W0 (Ω)]d and with (Lp(·) (Ω), k · kp(·) ) we denote all the spaces Lp(·) (Ω), [Lp(·) (Ω)]d and [Lp(·) (Ω)]nd with their norms. Furthermore, when it is clear from the context, we omit the dot · to denote the inner product between two vectors of Rd or of Rnd . L,p(·)

A weak solution of (2.1.1) is a function u ∈ W0 L,p(·) critical point of I : W0 (Ω) → R defined by Z I(u) = M (IL (u)) − F (x, u)dx, Ω

(Ω) which is a

(2.2.1)

20

Chapter 2. p(x)-Polyharmonic Elliptic Kirchhoff Systems L,p(·)

where IL (u) is given in (2.1.2). The functional I is of class C 1 (W0 (Ω)) L,p(·) and u ∈ W0 (Ω) is a critical point of I if Z Z p(x)−2 M (IL (u)) |DL u| DL uDL φ dx − f (x, u)φ dx = 0 (2.2.2) Ω

Ω

L,p(·)

for all φ ∈ W0 (Ω). We point out that the operation between DL u and DL φ is the d-Euclidean inner product if L is it is the ndR even, while p(x)−2 Euclidean inner product if L is odd. Moreover, Ω |DL u| DL uDL φ dx is the p(x)-polyharmonic operator ∆Lp(x) in the weak sense. We prove the existence of an unbounded sequence of weak solutions of (2.1.1), by using the symmetric mountain pass Theorem 2.2.3. Lemma 2.2.4 (Lemma 3 of [42]). Let ℘ be an exponent ℘ > 1 and X, (Xk )k ⊂ Rs be such that (|Xk |℘−2 Xk − |X|℘−2 X) · (Xk − X) → 0,

as k → ∞.

Then Xk → X as k → ∞. Proof. First, we claim that (Xk )k is bounded. Indeed, suppose by contradiction that (Xk )k is unbounded, then there exists a subsequence, still denoted by (Xk )k , such that |Xk | → ∞. On the other hand, (|Xk |℘−2 Xk − |X|℘−2 X) · (Xk − X) ∼ |Xk |℘ , hence also (|Xk |℘−2 Xk − |X|℘−2 X) · (Xk − X) → ∞ when k → ∞, which is absurd. This proves the claim. Fix now a subsequence (Xkj )j of (Xk )k . Clearly, also (Xkj )j is bounded. Therefore, there exists a subsequence of (Xkj )j , still denoted with (Xkj )j such that Xkj → α for some α ∈ Rs . This implies that (|α|℘−2 α − |X|℘−2 X) · (α − X) = 0, that is α = X. Noting that this is true for every subsequence of (Xkj )j , we obtain that Xk → X as k → ∞. 2 L,p(·)

Let (W ? , k · k? ) denote the dual space of W0 (Ω) and h·, ·i stand for L,p(·) ? the duality pairing between W and W0 (Ω). The following property holds.

21 L,p(·)

Lemma 2.2.5. IL0 : W0 (Ω) → W ? has the (S+ ) property, i.e. if L,p(·) L,p(·) (uk )k ⊂ W0 (Ω) converges weakly to u in W0 (Ω) and lim suphIL0 (uk ), uk − ui ≤ 0,

(2.2.3)

k→∞

L,p(·)

then (uk )k converges strongly to u in W0

(Ω). L,p(·)

Proof. Clearly the functional IL is of class C 1 (W0 (Ω)) and convex, so that IL (u) ≤ lim inf k→∞ IL (uk ) by the weak lower semicontinuity of L,p(·) IL in W0 (Ω). By the convexity of IL for all k Z IL (u) + |DL uk |p(x)−2 DL uk DL (uk − u)dx ≥ IL (uk ), Ω

so that IL (u) ≥ lim supk→∞ IL (uk ) by (2.2.3). In conclusion, lim IL (uk ) = IL (u).

k→∞

(2.2.4) L,p(·)

Furthermore, by (2.2.3), (2.2.4) and the fact that uk * u in W0 (Ω), we get Z (|DL uk |p(x)−2 DL uk − |DL u|p(x)−2 DL u)DL (uk − u)dx → 0 as k → ∞. Ω

Hence, (|DL uk |p(x)−2 DL uk − |DL u|p(x)−2 DL u)DL (uk − u) → 0 in L1 (Ω), being (|DL uk |p(x)−2 DL uk − |DL u|p(x)−2 DL u)DL (uk − u) ≥ 0 by convexity. Thus, up to a subsequence, (|DL uk |p(x)−2 DL uk −|DL u|p(x)−2 DL u)DL (uk −u) → 0 a.e. in Ω. (2.2.5) Fix x ∈ Ω for which (2.2.5) holds and apply Lemma 2.2.4 with s = 1, if L is even and with s = n if L is odd, with ℘ = p(x), Xk = DL uk (x) and X = DL u(x). Hence, Lemma 2.2.4 implies that DL uk converges to DL u a.e. in Ω, and in turn |DL uk |p(x) converges to |DL u|p(x) a.e. in Ω. Consider the sequence (gk )k in L1 (Ω) defined pointwise by ( p(x) ) 1 |DL uk (x)|p(x) + |DL u(x)|p(x) DL uk (x) − DL u(x) gk (x) = − . p(x) 2 2

22


By convexity gk ≥ 0 and gk (x) → |DL u(x)|p(x) /p(x) for a.a. x ∈ Ω. Therefore, by the Fatou lemma and (2.2.4) we have p(x) Z Z 1 DL uk − DL u IL (u) ≤ lim inf gk dx = IL (u) − lim sup dx k→∞ 2 k→∞ Ω Ω p(x) 1 ≤ IL (u) − lim sup ρp(·) (DL uk − DL u). p+ 2p+ k→∞ Hence, lim supk→∞ ρp(·) (DL uk − DL u) = 0, that is limk→∞ kuk − uk = 0 by Lemma 1.1.4, as required. 2 For a somehow less direct proof of the previous lemma we refer to Theorem 3.1 of [50]. Let us end the section with some remarks on the main structural assumptions on M and f . First, condition (M ) implies in particular that M (τ ) > 0 for all τ > 0. Furthermore, (M ) ensures also that M (τ ) ≥ M (1)τ γ

for all τ ∈ [0, 1]

(2.2.6)

and, similarly, for all ε > 0 M (τ ) ≤ δτ γ

for all τ ≥ ε,

(2.2.7)

where δ = δ(ε) = M (ε)/εγ > 0. Property (2.2.7) hence yields lim τ −µ/p+ M (τ ) = 0,

τ →∞

(2.2.8)

being µ > γp+ by (f3 ). We note in passing that conditions (f2 ) and (f3 ) force that µ ≤ q− . An example of function f : R → R verifying (f1 )–(f4 ), when d = 1, p ≡ Const. and q ≡ Const. is given by ( |s|q−2 s − |s|℘−2 s, if |s| ≤ 1, f (s) = |s|q−2 s − |s|p−2 s, if |s| ≥ 1, for all ℘, p and q such that 1 < γp < ℘ and γp < q < p∗L , with γ ∈ [1, pL ). Indeed, (f1 ), (f2 ) and (f4 ) hold immediately with C = 2, being γp < max{℘, q}. Moreover, it can be easily proved that for all µ ∈ (γp, q] there exist U = max{(q/p)1/(q−p) , [µ(℘ − p)/℘(µ − p)]1/p } > 1 and c = µ(℘ − p)/℘p such that (f3 ) is valid.

23

2.3

An Unbounded Sequence of Weak Solutions

Lemma 2.3.1. The energy functional I satisfies the (PS) condition. L,p(·)

Proof. Let (uk )k ⊂ W0

(Ω) be a (PS) sequence for I, that is

sup |I(uk )| = A < ∞ and I 0 (uk ) → 0 as k → ∞.

(2.3.1)

k

Two possible cases arise: either inf kuk k = d > 0 or inf kuk k = 0, so that k

k

we divide the proof into two parts. Case inf kuk k = d > 0. We begin with proving that (uk )k is bounded. k

Observe that, by (1.1.4) we get for all k ∈ N Z |DL uk |p(x) 1 1 IL (uk ) = dx ≥ ρp(·) (DL uk ) ≥ min{dp+,dp− }. (2.3.2) p(x) p p + + Ω Denote by κ = κ(d) the number corresponding to σ = min{dp+ , dp− }/p+ > 0 in (M ), so that M (IL (uk )) ≥ κ > 0 for all k.

(2.3.3)

Hence, by (2.2.1), (2.2.2), (M ) and (f3 ) we obtain Z 1 0 I(uk )− hI (uk ), uk i = M (IL (uk )) − F (x, uk )dx µ Z ZΩ 1 p(x) − M (IL (uk )) |DL uk | dx− uk f (x, uk )dx µ Ω Ω 1 p+ ≥ − M (IL (uk ))IL (uk ) γ µ Z 1 − F (x, uk ) − uk f (x, uk ) dx µ Ω + Z 1 ≥ Cd IL (uk ) − F (x, uk ) − uk f (x, uk ) dx, µ Ωk 1 p+ where Cd = − κ > 0 and Ωk = {x ∈ Ω : |uk (x)| ≤ U }. Note γ µ that dU = supx∈Ω, |s|≤U [F (x, s) − sf (x, s)/µ]+ < ∞. Indeed, by (f2 ) and

24


(f3 ), for all x ∈ Ω and |s| ≤ U + sf (x, s) sf (x, s) F (x, s) − ≤ F (x, s) − µ µ Z 1 C d F (x, ts)dt + |s| + |s|q(x) ≤ µ 0 dt Z 1 C |f (x, ts)|·|s|dt + ≤ |s| + |s|q(x) µ 0 Z 1 C q(x)−1 q(x) ≤C (|s| + t |s| )dt + |s| + |s|q(x) µ 0 C 1 |s|q(x) + |s| + |s|q(x) ≤ C |s| + q(x) µ 1 ≤C 1+ (U + U q(x) ) < ∞. µ Therefore 1 I(uk )− hI 0 (uk ), uk i ≥ Cd IL (uk ) − D, (2.3.4) µ where D = dU |Ω| < ∞. By the fact that I 0 (uk ) ∈ W ? and by (2.3.1), we have 1 1 I(uk ) − hI 0 (uk ), uk i ≤ A + |hI 0 (uk ), uk i| ≤ A + Bkuk k, (2.3.5) µ µ where µB = supk kI 0 (uk )k? . Now, combining together (2.3.4) and (2.3.5) and using (1.1.4), we get Cd Cd A + Bkuk k ≥ ρp(·) (DL uk ) − D ≥ min{kuk kp+ , kuk kp− } − D. p+ p+ L,p(·)

(Ω). Hence, being 1 < p− ≤ p+ , the sequence (uk )k is bounded in W0 L,p(·) L,p(·) (Ω) such that, up (Ω) is reflexive, there exists u ∈ W0 Since W0 L,p(·) (Ω). Next we to a subsequence, (uk )k converges weakly to u ∈ W0 show that actually that convergence is strong. Indeed, for all k Z 0 hI (uk ), uk − ui = M (IL (uk )) |DL uk |p(x)−2 DL uk DL (uk − u)dx Ω Z − f (x, uk )(uk − u)dx. Ω

(2.3.6)

25

By (2.3.1), as k → ∞ |hI 0 (uk ), uk − ui| ≤ kI 0 (uk )k? kuk − uk ≤ c0 kI 0 (uk )k? → 0,

(2.3.7) L,p(·)

where c0 = supk kuk − uk < ∞, since (uk )k is bounded in W0 (Ω). L,p(·) q(·) By (f2 ) the embedding W0 (Ω) ,→ L (Ω) is compact. Hence, (uk )k q(·) converges strongly to u in L (Ω). Denote by d1 = supk kuk kq(·) < ∞. By (f2 ), (f3 ), Theorem 1.1.1 and (1.1.2) with q in place of p, letting k → ∞ we have Z f (x, uk )(uk − u)dx Ω Z ≤ C (1 + |uk |q(x)−1 )|uk − u|dx Ω Z (2.3.8) q(x)−1 |uk − u|dx = C kuk − uk1 + |uk | Ω ≤ C 1 + |Ω| + cq k|uk |q(·)−1 kq0 (·) kuk − ukq(·) ≤ C1 kuk − ukq(·) → 0, where cq = 1 + 1/q− − 1/q+ is the constant which appears in Hölder’s inequality, q 0 is the conjugate exponent of q and q −1

q −1

C1 = 3C max{1, |Ω|, cq d1− , cq d1+ } < ∞, q −1

q −1

− + being k|uk |q(·)−1 kq0 (·) ≤ max{kuk kq(·) , kuk kq(·) } for all k, by Lemma 1.1.5. Consequently, by (2.3.6)–(2.3.8) we obtain Z M (IL (uk )) |DL uk |p(x)−2 DL uk DL (uk − u)dx → 0 as k → ∞,

Ω

that is Z

|DL uk |p(x)−2 DL uk DL (uk − u)dx → 0 as k → ∞

(2.3.9)

Ω

by (2.3.3), namely limk→∞ hIL0 (uk ), uk − ui = 0. The conclusion follows at once by Lemma 2.2.5. Case inf kuk k = 0. Either 0 is an accumulation point for the real k

sequence (kuk k)k , and so there is a subsequence of (uk )k strongly converging to u = 0, or 0 is an isolated point of the sequence (kuk k)k , and so

26


there is a subsequence, denoted by (kuj k)j , such that inf j kuj k = d > 0. In the first case we are done, while in the latter case we can proceed as in Step 1, starting now from the subsequence (uj )j . 2 Lemma 2.3.2. There exist r, α > 0 such that I(u) ≥ α for every u ∈ L,p(·) W0 (Ω), with kuk = r. Proof. By (f4 ) for all ε > 0 there exists δ = δ(ε) > 0 such that |F (x, s)| ≤ ε|s|γp+ for x ∈ Ω and |s| < δ. On the other hand, by (f2 ) and (f3 ) for x ∈ Ω and |s| ≥ δ we get Z 1 Z 1 d F (x, ts) dt ≤ |F (x, s)| ≤ |f (x, ts)|·|s|dt dt 0 0 Z 1 |s|q(x) q(x)−1 q(x) (|s| + t |s| )dt ≤ C |s| + ≤ cε |s|q(x) , ≤C q(x) 0 where cε = 2C max {δ 1−q+ , δ 1−q− , 1/q− } > 0. Hence, we have shown that for all ε > 0 there exists cε > 0 such that |F (x, s)| ≤ ε|s|γp+ + cε |s|q(x)

(2.3.10)

for all x ∈ Ω and s ∈ Rd . p+ −γ Let ε ∈ (0, M (1)(p+ Sγp ) ) be fixed, where Sγp+ is the Sobolev + γp+ >0 constant given in (1.1.8) when h+ = γp+ . Put κε = M (1)/pγ+ −εSγp + q+ q− and Cε = cε max{Sq+ , Sq+ }, where Sq+ is the Sobolev constant given in (1.1.8) when h+ = q+ . Take r ∈ (0, 1] so small that rq− −γp+ < κε /Cε . L,p(·) Then, for all u ∈ W0 (Ω), with kuk = r, by (2.2.1), (2.3.10), (2.2.6) and (1.1.4) we obtain + I(u) ≥ M (IL (u)) − εkukγp γp+ − cε ρq(·) (u) γ M (1) + ≥ ρp(·) (DL u) − εkukγp γ γp+ − cε ρq(·) (u) p+ M (1) q+ q− + min{kukγp+ , kukγp− } − εkukγp ≥ γp+ − cε max{kukq(·) , kukq(·) } pγ+ M (1) γp+ γp+ γp+ − cε max{Sqq++ , Sqq+− }rq− ≥ r − εSγp r + pγ+ ≥ rγp+ (κε − Cε rq− −γp+ ).

27

Hence, I(u) ≥ α, where α = rγp+ (κε − Cε rq− −γp+ ) > 0 for all u ∈ L,p(·) W0 (Ω), with kuk = r, by construction. 2 L,p(·)

Lemma 2.3.3. For every finite dimensional subspace W of W0 there exists R0 = R0 (W ) > 0 such that

(Ω)

I(u) ≤ 0 for all u ∈ W \ BR0 (W ), where BR0 (W ) = {u ∈ W : kuk < R0 }. Proof. We begin with proving that there exist two constant dµ , d0 > 0 such that F (x, s) ≥ dµ |s|µ − d0

for all x ∈ Ω and s ∈ Rd .

(2.3.11)

Indeed, taking inspiration from [34], we define S = {s ∈ Rd : |s| = U } and for all x ∈ Ω and s ∈ S we consider the function gs : [1, ∞) → R, such that gs (t) = F (x, t1/µ s) for all t ∈ [1, ∞). Now, by (f3 ) gs0 (t) =

t1/µ sf (x, t1/µ s) F (x, t1/µ s) gs (t) ≥ = . µt t t

An integration on [1, t] gives gs (t) ≥ tgs (1) ≥ cS t,

(2.3.12)

where cS = inf{F (x, s) : x ∈ Ω, s ∈ S} ≥ c > 0, by (f3 ). Let s ∈ Rd be −1/µ such that |s| > U and consider r = t0 s, where t0 = (|s|/U )µ , so that r ∈ S. For all x ∈ Ω and s ∈ Rd , with |s| ≥ U , we get by (2.3.12) 1/µ

F (x, s) = F (x, t0 r) = gr (t0 ) ≥ cS t0 = dµ |s|µ ,

(2.3.13)

where dµ = cS /U µ . Now, fix ε > 0. By (2.3.10) for all x ∈ Ω and s ∈ Rd such that |s| ≤ U , we get |F (x, s) − dµ |s|µ | ≤ |F (x, s)| + dµ |s|µ ≤ ε|s|γp+ + cε |s|q(x) + dµ |s|µ ≤ εU γp+ + cε U q(x) + dµ U µ < ∞

28


Hence, for all x ∈ Ω and s ∈ Rd , |s| ≤ U d0 =

|F (x, s) − dµ |u|µ | ≥ −F (x, s) + dµ |u|µ .

sup

(2.3.14)

x∈Ω,|s|≤U

Therefore, combining the two estimates (2.3.13) and (2.3.14) we have (2.3.11). L,p(·) Let W be a finite dimensional subspace of W0 (Ω) and let u ∈ W with kuk = 1 be fixed. For all t ≥ 1 it results IL (tu) ≤

tp+ ρp(·) (DL u) < tp+ , p−

by (1.1.4) and (2.1.2). Hence, by (2.3.11), (M ) and the fact that M is nondecreasing, we have for all t ≥ 1 I(tu) ≤ M (IL (tu)) − dµ tµ kukµµ + d0 |Ω| ≤ t−µ M (tp+ ) − Dµ tµ + d0 |Ω|, where Dµ = dµ cµW > 0 and cW is such that kwkµ ≥ cW kwk for all w ∈ W . Now, fix ε ∈ (0, Dµ ), so that by (2.2.8) there exists t0 ≥ 1 such that t−µ M (tp+ ) ≤ ε for all t ≥ t0 . Thus, I(tu) ≤ −βε tµ + d0 |Ω| → −∞

as t → ∞,

being βε = Dµ − ε > 0. Therefore, as R → ∞ sup kuk=R,u∈W

I(u) =

sup

I(Ru) → −∞.

kuk=1,u∈W

Hence, there exists R0 > 0 so large that I(u) ≤ 0 for all u ∈ W , with kuk = R, and R ≥ R0 . This completes the proof. 2 Theorem 2.3.4. Problem (2.1.1) admits an unbounded sequence of weak solutions. Proof. By (f1 ) we know that F (x, ·) is even in R, so that also I is even. Since I(0) = 0, by Lemmas 2.3.1– 2.3.3 and Theorem 2.2.3 we deduce the existence of an unbounded sequence of weak solutions to problem (2.1.1). 2

Chapter 3

Multiplicity of Solutions for Eigenvalue p-Polyharmonic Kirchhoff Systems In this chapter we establish the existence of two nontrivial weak solutions of some eigenvalue problems. These results complete in several directions some recent papers, where a Ricceri type critical point theorem was used. The main arguments of this chapter are based on two recent theorems due to Arcoya and Carmona in [7], which we use in a slightly modified version, see Section 3.2. The results of this chapter contain and extend the ones of [33].

3.1

Introduction

The three critical points theorems due to Pucci and Serrin [93, 94] and to Ricceri [100] have been applied with success in several problems involving differential equations with one real parameter. Recently, Arcoya and Carmona in [7] extended Pucci and Serrin theorem to a wide class of continuous functionals not necessarily differentiable. For applications we refer to the recent book by Kristály, R˘adulescu and Varga [71] and the references therein. See also [22, 28, 44, 69, 70, 75, 102]. In Section 3.3 we adopt these techniques in order to study the follow-

30

Chapter 3. Eigenvalue Systems

ing higher order eigenvalue Kirchhoff system, for λ ∈ R ( M (kukp )∆Lp u = λ{w(x)|u|p−2 u + f (x, u)} in Ω, Dα u(x) ∂Ω = 0 for all α, with |α| ≤ L − 1,

(3.1.1)

where L = 1, 2, . . . , Ω ⊂ Rn is a bounded domain, n ≥ 1, u = (u1 , . . . , ud ) = u(x), d ≥ 1, p > 1 and α is a multi-index. For every u = (u1 , . . . , ud ) ∈ [W0L,p (Ω)]d , the vectorial p-polyharmonic operator ∆Lp u has to be understood as the d-vector defined in (2.1.3) with p(x) ≡ p The perturbation f : Ω × Rd → Rd is a Carathéodory (p − 1)-sublinear function; for the exact conditions on f we refer to Section 3.3. The weight w is assumed positive a.e. in Ω and of class L$ (Ω), with $ > + max{1, n/Lp}. We assume that the Kirchhoff function M : R+ 0 → R0 is of the special type M (τ ) = a + bγτ γ−1 ,

a > 0,

b ≥ 0, γ ≥ 1,

(3.1.2)

and R τ we denote by M its primitive which vanishes at zero, namely M (τ ) = M (z)dz. Hence, in Section 3.3 we are dealing only with the non0 degenerate case in which M is strictly positive. By using a slightly modified version of Theorem 3.4 of [7], in Theorem 3.3.7 we determine precisely the intervals of λ’s for which problem (3.1.1) admits only the trivial solution and for which (3.1.1) has at least two nontrivial solutions. In Section 3.4, in the complements to the chapter, we analyze some divergence type eigenvalue equations under Dirichlet or under Robin boundary conditions. The one parameter problem ( −divA(x, Du) = λ h(x, u) in Ω, (3.1.3) u=0 on ∂Ω, where Ω ⊂ Rn is a bounded domain, n ≥ 2, and the nonlinearities A : Ω × Rn → Rn and h : Ω × R → R fulfill certain natural structural conditions, is widely studied, see e.g. [24, 37, 43, 106]. In these papers the authors use several variational techniques, in order to establish existence and multiplicity of solutions for (3.1.3).

31

Following this trend, for λ ∈ R, we first study the problem ( −divA(x, Du) = λ{w(x)|u|p−2 u + f (x, u)} in Ω, u=0 on ∂Ω,

(3.1.4)

where Ω is a bounded domain of Rn , n ≥ 1, p > 1. The perturbation f : Ω × R → R is a Carathéodory function, essentially of the type studied in Section 3.3 when d = 1 and the weight w is assumed positive a.e. in Ω and of class L$ (Ω), with $ > max{1, n/p}. Then we consider, for λ, µ ∈ R, the inhomogeneous Robin problem ( −divA(x, Du) = λ{w(x)|u|p−2 u + f (x, u)} in Ω, (3.1.5) A(x, Du) · ν + w1 (x)|u|p−2 u = λµg(x, u) on ∂Ω, where Ω ⊂ Rn is now a bounded domain of class C 1 , ν is the outward normal vector field on ∂Ω and 1 < p < n, n ≥ 2. Moreover, we endow ∂Ω with the (n − 1)-dimensional Lebesgue measure. The perturbation f is assumed to be as in model (3.1.4), while the weight w is assumed positive a.e. in Ω and of class L$ (Ω), with $ > n/p. The weight w1 is in L(n−1)/(p−1) (∂Ω) and there exists a positive constant w0 such that w1 (x) ≥ w0 > 0 for all x ∈ ∂Ω. Also for these two divergence type eigenvalue problems we find out the intervals of λ’s for which there exists only the trivial solution and for which there are at least two nontrivial solutions. We want to stress here that these intervals are expressed in terms of the first eigenvalue of the main operator of the problem, which we need to be positive. This is the main reason that led us to set all the problems of this chapter in the classical Sobolev spaces. Indeed, in the setting of variable exponent Sobolev spaces may arise that, for instance, the first eigenvalue of the p(x)-Laplacian is zero. In [51], Fan et al. prove that the first eigenvalue of the p(x)-Laplacian is positive if the function p satisfies some restrictive monotonicity properties. However, as far as we know, no sufficient conditions have been given in literature for higher order p(x)-operators.

3.2

Preliminaries and Auxiliary Results

Let us now introduce the main notation and assumptions for Theorem 3.2.1. Let (X, k · k) be a reflexive real Banach space, with (topolog-

32


ical) dual space (X ? , k · k? ). Assume that Φ and Ψ are two functionals on X, verifying the following hypotheses. (H1 ) Φ and Ψ are weakly lower semicontinuous and continuously Gâteaux differentiable in X, and Ψ is nonconstant; (H2 ) Φ0 : X → X ? has the (S+ ) property; (H3 ) Ψ0 : X → X ? is a compact operator; (H4 ) there exists an interval I ⊂ R such that the one parameter family of functionals Jλ = Φ + λΨ, λ ∈ I, is coercive in X, i.e. for all λ∈I lim Jλ (u) = ∞.

kuk→∞

By convenience, for every r ∈ (inf u∈X Ψ(u), supu∈X Ψ(u)) we introduce the two functions inf

ϕ1 (r) =

inf

v∈Ψ−1 (r)

Ψ(u) − r

u∈Ψ−1 (Ir )

inf

ϕ2 (r) =

Φ(v) − Φ(u)

v∈Ψ−1 (r)

sup u∈Ψ−1 (I r )

,

Ir = (−∞, r).

(3.2.1)

I r = (r, ∞).

(3.2.2)

Φ(v) − Φ(u)

Ψ(u) − r

,

The next result is a slight variant of the differentiable version of the Arcoya and Carmona Theorem 3.4 in [7]. Theorem 3.2.1. Let Φ, Ψ : X → R be two functionals satisfying (H1 )– (H4 ) and suppose that there exists r ∈ inf Ψ(u), sup Ψ(u) such that ϕ1 (r) < ϕ2 (r), (3.2.3) u∈X

u∈X

then the following properties hold. (i) The functional Jλ admits at least one critical point for every λ ∈ I. (ii) If furthermore (ϕ1 (r), ϕ2 (r)) ∩ I 6= ∅, then (a) Jλ has at least three critical points for every λ ∈ (ϕ1 (r),ϕ2 (r))∩I.

33

(b) Jϕ1 (r) has at least two critical points, provided that ϕ1 (r) ∈ I. (c) Jϕ2 (r) has at least two critical points, provided that ϕ2 (r) ∈ I. Proof. If we prove that the functional Jλ satisfies the (PS) condition for all λ ∈ I, then we apply Theorem 3.4 of [7] and we conclude. Let (uk )k ⊂ X be a (PS) sequence for Jλ , i.e. (Jλ (uk ))k is bounded and Jλ0 (uk ) → 0.

(3.2.4)

Clearly (uk )k is bounded in X, since Jλ is coercive in X for all λ ∈ I by (H4 ). Therefore, up to a subsequence, still denoted by (uk )k , we get that uk * u weakly in X, being X reflexive. Consequently, Ψ0 (uk ) → Ψ0 (u) in X ? by (H3 ). Hence, hΦ0 (uk ), uk − ui = hJλ0 (uk ), uk − ui − λhΨ0 (uk ), uk − ui = hJλ0 (uk ), uk − ui − λhΨ0 (uk ) − Ψ0 (u), uk − ui − λhΨ0 (u), uk − ui ≤ kJλ0 (uk )k? kuk − uk + |λ| · kΨ0 (uk ) − Ψ0 (u)k? kuk − uk + |λ| · |hΨ0 (u), uk − ui|, where, h·, ·i denotes the duality pairing between X and its dual space X ? . Therefore, by (3.2.4) and the facts that Ψ0 (uk ) → Ψ0 (u) in X 0 , uk * u weakly in X, it results that lim suphΦ0 (uk ), uk − ui ≤ 0. k→∞

Assumption (H2 ) implies that uk → u strongly in X. This completes the proof of the theorem. 2 We introduce now also a variant of the differentiable form of the Arcoya and Carmona Theorem 3.10 in [7], which will be useful in Section 3.4. Theorem 3.2.2. Let Φ, Ψ : X → R be two functionals satisfying (H1 )– (H3 ) and let Υ : X → R be continuously Gâteaux differentiable with compact derivative Υ0 . Furthermore, assume the following condition. (H40 ) There exists an interval I ⊂ R and a number η > 0, such that for every λ ∈ I and every µ ∈ [−η, η] the functional Jλ,µ = Φ + λ(Ψ + µΥ) is coercive in X.

34


If (3.2.3) holds and (ϕ1 (r), ϕ2 (r))∩I 6= ∅, then for every compact interval [a, b] ⊂ (ϕ1 (r), ϕ2 (r)) ∩ I there exists δ ∈ (0, η) such that if |µ| < δ, the functional Jλ,µ admits at least three critical points for every λ ∈ [a, b]. Proof. As in the proof of Theorem 3.2.1, it is possible to show that for all λ ∈ I and µ ∈ [−η, η] the functional Jλ,µ satisfies the (PS) condition. The conclusion of the theorem follows as a consequence of Theorem 3.10 of [7]. 2 Throughout the chapter, Ls (Ω), s ≥ 1, denotes the standard Lebesgue space, endowed with the canonical norm k · ks , and s0 is the conjugate exponent of s. Moreover, for brevity in notation, Ls (A; ω), s ≥ 1, denotes the weighted Lebesgue space equipped with the norm Z 1/s s kuks,ω,A = ω(x)|u(x)| dx . A

When A = Ω we denote kuks,ω,A simply by kuks,ω . Let us state in this section a useful result for general weighted Lebesgue spaces, which is well-known in the framework of the standard Lebesgue spaces. The proof is left to the reader, since it is standard, see also [98]. Lemma 3.2.3. Let A be a measurable subset of RN , N ≥ 1, ω be a weight on A and s ∈ [1, ∞). If (uk )k and u are in Ls (A; ω) and uk → u in Ls (A; ω) as k → ∞, then there exist a subsequence (ukj )j of (uk )k and a function h ∈ Ls (A; ω) such that a.e. in A (i) ukj → u as j → ∞;

(ii) |ukj | ≤ h for all j ∈ N.

Furthermore, in the whole chapter p∗L denote the Sobolev critical exponent   np , if n > Lp, ∗ pL = n − Lp ∞, if 1 ≤ n ≤ Lp. If L = 1 we simply write p∗ in place of p∗1 .

3.3

Existence of Three Solutions

In this section, X = [W0L,p (Ω)]d and X ? = {[W0L,p (Ω)]d }? , which in what follows will be simply denoted by W ? . Since we are in the vectorial

35

setting, we define the norm for every function u in the solution space [W0L,p (Ω)]d as kuk = k |DL u|s kp , where DL u = (DL u1 , . . . , DL ud ) is a d-vector if L is even, otherwise it is an nd-vector and DL ui for i = 1, . . . , d is defined in (1.2.2). Moreover, | · |s is the s-Euclidean norm, with s = d if L is even, s = nd if L is odd. For simplicity in notation, throughout this section we drop the exponents d and nd in all the functional spaces involved in the treatment. Furthermore, we simply denote by the same symbol | · | the Euclidean norm in the spaces Rd and Rnd . Let λ1 be the first eigenvalue of the problem ( in Ω, ∆Lp u = λ w(x)|u|p−2 u (3.3.1) α D u ∂Ω = 0 for all α, with |α| ≤ L − 1, in W0L,p (Ω), that is λ1 is defined by the Rayleigh quotient R |D u|p dx RΩ L λ1 = inf . p u∈W0L,p (Ω) Ω w(x)|u| dx

(3.3.2)

u6=0

Clearly, λ1 kukpp,w ≤ kukp

for all u ∈ W0L,p (Ω).

(3.3.3)

Before introducing the following proposition we remark that the embedding W0L,p (Ω) ,→ Lp (Ω; w) is compact. Indeed, if n > Lp, then the 0 embedding W0L,p (Ω) ,→ L$ p (Ω) is compact, since $ > n/Lp implies that 0 $0 p < p∗L . Moreover, L$ p (Ω) ,→ Lp (Ω; w) continuously, by Hölder’s in0 equality kukpp,w ≤ kwk$ kukp$0p for all u ∈ L$ p (Ω), being w ∈ L$ (Ω). 0 Analogously, if n ≤ Lp, then W0L,p (Ω) ,→ L$ p (Ω) is compact, being 0 $0 p < ∞. Also in this case the embedding L$ p (Ω) ,→ Lp (Ω; w) is continuous. Hence, in both cases we can conclude that the embedding W0L,p (Ω) ,→ Lp (Ω; w) is compact. In particular, let us denote by Sp,w > 0 the best constant such that kukp,w ≤ Sp,w kuk for all u ∈ W0L,p (Ω).

(3.3.4)

36


Proposition 3.3.1. The infimum λ1 in (3.3.2) is attained and p λ1 = 1/Sp,w > 0.

Proof. First, I(u) = kukp and J (u) = kukpp,w are continuously Fréchet differentiable, convex in W0L,p (Ω), and clearly I 0 (0) = J 0 (0) = 0. Moreover, J 0 (u) = 0 implies u = 0. In particular, I and J are weakly lower semicontinuous on W0L,p (Ω). Actually, J is weakly sequentially continuous on W0L,p (Ω). Indeed, if (uk )k ⊂ W0L,p (Ω) and uk * u in W0L,p (Ω), then uk → u in Lp (Ω; w), being the natural embedding between W0L,p (Ω) and Lp (Ω; w) compact. This implies at once that J (uk ) = kuk kpp,w → J (u) = kukpp,w , as claimed. Now, two possible cases arise: either W = {u ∈ W0L,p (Ω) : J (u) = kukpp,w ≤ 1} is bounded with respect to the norm k · k, or W is unbounded. In the first case we need no more assumptions, in the second case, since I is coercive in W0L,p (Ω), it is coercive also in the subset W . In conclusion, all the assumptions of Theorem 6.3.2 of [18] are fulfilled and so λ1 is attained in W0L,p (Ω). p p Finally, J (u) = kukpp,w ≤ Sp,w kukp = Sp,w I(u) for all u ∈ W0L,p (Ω), p with Sp,w given in (3.3.4). Hence, λ1 ≥ 1/Sp,w > 0 by (3.3.2). Since Sp,w p . 2 is the best constant for (3.3.4), we get λ1 = 1/Sp,w Denote by u1 ∈ W0L,p (Ω) the normalized eigenfunction corresponding to λ1 , that is ku1 kp,w = 1 and λ1 = ku1 kp . On f we assume the next condition. (F) Let f : Ω × Rd → Rd be a Carathéodory function, f 6≡ 0, satisfying the following properties. (a) There exist two measurable functions f0 , f1 : Ω → R and an exponent q ∈ (1, p), such that 0 ≤ f0 (x) ≤ Cf w(x), 0 ≤ f1 (x) ≤ Cf w(x) a.e. in Ω for some appropriate constant Cf > 0, and |f (x, s)| ≤ f0 (x) + f1 (x)|s|q−1 for a.a. x ∈ Ω and all s ∈ Rd . (b) There exists p? ∈ (p, p∗L /$0 ) such that lim sup s→0

uniformly a.e. in Ω.

|f (x, s)| < ∞, w(x)|s|p? −1

37

1 M (λ1 ) (c) F (x, u1 )dx > − 1 , where f (x, s) = Ds F (x, s), p aλ1 Ω F is such that F (x, 0) = 0, the constant a is given in (3.1.2) and u1 is the first normalized eigenfunction defined above. Z

Note that, in the more familiar and standard setting in the literature in which w ∈ L∞ (Ω), the exponent p? in (F)–(b) belongs to the open interval (p, p∗L ). Furthermore, in (F)–(c) we can have R M (λ1 ) = aλ1 , when b = 0. In that case, (F)–(c) simply reduces to Ω F (x, u1 )dx > 0. Condition (F) implies that f (x, 0) = 0 for a.a. x ∈ Ω. Otherwise, there exists A ⊂ Ω, |A| > 0, such that |f (x, 0)| > 0 and w(x) > 0 ? for all x ∈ A. Hence, lims→0 |f (x, s)|/w(x)|s|p −1 = ∞ for all x ∈ A, contradicting (F)–(b). Furthermore, lim sup s→0

|F (x, s)| < ∞ uniformly a.e. in Ω, w(x)|s|p?

(3.3.5)

by the L’Hôpital rule. Finally, |f (x, s)| ∈ R+ . p−1 w(x)|s| s6=0,x∈Ω

Sf = ess sup

(3.3.6)

Indeed, Sf > 0 being f ≡ 6 0 and Sf < ∞, since first by (F)–(b) |f (x, s)| |f (x, s)| ? lim = lim |s|p −p = 0 uniformly a.e. in Ω, ? s→0 w(x)|s|p−1 s→0 w(x)|s|p −1 being p? > p. Moreover, |f (x, s)|/w(x)|s|p−1 ≤ 2Cf |s|q−p for a.a. x ∈ Ω and all |s| ≥ 1 by (F)–(a), that is |f (x, s)| = 0 uniformly a.e. in Ω, s→∞ w(x)|s|p−1 lim

since q < p. Hence the positive number λ? =

a λ1 1 + Sf

(3.3.7)

38


is well defined. Furthermore, by (3.3.6) |F (x, s)| Sf . = p p s6=0,x∈Ω w(x)|s|

ess sup

(3.3.8)

In what follows we define two functionals Φ and Ψ and we prove in four lemmas that they satisfy conditions (H1 )–(H4 ) required in the main Theorem 3.2.1. Lemma 3.3.2. The functional 1 Φ(u) = M (kukp ), p

Φ : W0L,p (Ω) → R,

is convex, weakly lower semicontinuous and of class C 1 (W0L,p (Ω)). Moreover, Φ0 : W0L,p (Ω) → W ? verifies the (S+ ) condition, i.e. for every sequence (uk )k ⊂ W0L,p (Ω) such that uk * u weakly in W0L,p (Ω) and Z p lim sup M (kuk k ) |DL uk |p−2 DL uk (DL uk − DL u)dx ≤ 0, (3.3.9) k→∞

Ω

it results that uk → u strongly in W0L,p (Ω). Proof. A simple calculation shows that Φ is convex in W0L,p (Ω), being M (kukp ) = akukp + bkukγp , p > 1 and γ ≥ 1. Moreover, we claim that Φ ∈ C 1 (W0L,p (Ω)). Indeed, Φ is Gâteaux differentiable in W0L,p (Ω) and for all u, v ∈ W0L,p (Ω) it results Z 0 p hΦ (u), vi = M (kuk ) |DL u|p−2 DL uDL vdx. Ω

Now, let u, (uk )

⊂ W0L,p (Ω) such that uk → u kΦ0 (uk ) − Φ0 (u)k? → 0 as

as k → ∞. We assert that k → ∞.

W0L,p (Ω),

Fix v ∈ with kvk = 1, and estimate Z p p−2 p p−2 M (ku k )|D u | D u −M (kuk )|D u| D u D vdx k L k L k L L L Ω Z M (kuk kp )|DL uk |p−2 DL uk ≤ (3.3.10) Ω −M (kukp )|DL u|p−2 DL u |DL v|dx

≤ M (kuk kp )|DL uk |p−2 DL uk − M (kukp )|DL u|p−2 DL u 0 p

39

where in the last step we have used Hölder’s inequality. Fix now a subsequence (ukj )j of (uk )k . Clearly, also ukj → u in W0L,p (Ω) and so DL ukj → DL u in Lp (Ω) as j → ∞. By Lemma 3.2.3, with A = Ω, N = n, s = p and ω ≡ 1, there exists a subsequence of (ukj )j , still denoted by (ukj )j verifying (i) and (ii), namely DL ukj → DL u

and

|DL ukj | ≤ h

for all j ∈ N and an appropriate function h ∈ Lp (Ω). Hence, 0 M (kuk kp )|DL uk |p−2 DL uk − M (kukp )|DL u|p−2 DL u p j j j n 0 0o p0 −1 p p−1 p p p−1 p ≤2 M (kukj k )|DL ukj | + M (kuk )|DL u| p0 0 ≤ 2p a + bγU p(γ−1) hp ∈ L1 (Ω), where U = supj kukj k < ∞, being (ukj )j convergent and so bounded in W0L,p (Ω). On the other hand, M (kuk kp )|DL uk |p−2 DL uk − M (kukp )|DL u|p−2 DL u → 0 as j → ∞, j j j since M is continuous. Thus, applying the Lebesgue dominated convergence theorem, we obtain that kM (kukj kp )|DL ukj |p−2 DL ukj − M (kukp )|DL u|p−2 DL ukp0 → 0 as j → ∞ which implies by (3.3.10) that Z M (kukj kp )|DL ukj |p−2 DL uk − M (kukp )|DL u|p−2 DL u DL vdx → 0 Ω

as j → ∞. Therefore we have proved that every subsequence (ukj )j has a subsequence, still denoted by (ukj )j , such that Z kΦ0 (ukj ) − Φ0 (u)k? = sup M (kukj kp )|DL ukj |p−2 DL ukj v∈W0L,p (Ω) kvk=1

Ω

−M (kukp )|DL u|p−2 DL u DL vdx → 0, as j → ∞. Hence, the whole sequence (Φ0 (uk ))k converges to Φ0 (u) in W ? and Φ is of class C 1 (W0L,p (Ω)). In particular Φ is weakly lower semicontinuous in W0L,p (Ω), by Corollary 3.9 of [23].

40


Let (uk )k ⊂ W0L,p (Ω) be such that uk * u weakly in W0L,p (Ω) and (3.3.9) holds. Then Z p lim M (kuk ) |DL u|p−2 DL uDL (uk − u)dx = 0, (3.3.11) k→∞

Ω 0

being |DL u|p−2 DL u ∈ Lp (Ω). Hence (3.3.9) is equivalent to Z lim sup M (kuk kp )|DL uk |p−2 DL uk k→∞ Ω −M (kukp )|DL u|p−2 DL u DL (uk − u)dx ≤ 0. Now, by convexity M (kuk kp )|DL uk |p−2 DL uk − M (kukp )|DL u|p−2 DL u DL (uk − u) ≥ 0, therefore Z

M (kuk kp )|DL uk |p−2 DL uk

lim

k→∞

Ω

−M (kukp )|DL u|p−2 DL u DL (uk − u)dx = 0. This implies by (3.3.11) Z lim M (kuk kp )|DL uk |p−2 DL uk DL (uk − u)dx = 0 k→∞

Ω

and in turn Z lim

k→∞

|DL uk |p−2 DL uk DL (uk − u)dx = 0,

(3.3.12)

Ω

being M (kuk kp ) ≥ a > 0 for all k ∈ N. On the other hand, by the weak lower semicontinuity of the norm, we get kDL ukpp ≤ lim inf kDL uk kpp k→∞

and by convexity Z p kDL ukp + p |DL uk |p−2 DL uk DL (uk − u)dx ≥ kDL uk kpp . Ω

(3.3.13)

(3.3.14)

41

Therefore, combining together (3.3.12) and (3.3.14), it results that kDL ukpp ≥ lim sup kDL uk kpp . k→∞

Taking into account (3.3.13), this gives lim kDL uk kpp = kDL ukpp .

k→∞

(3.3.15)

Consider the sequence (gk )k in L1 (Ω) defined pointwise by p |DL uk (x)|p + |DL u(x)|p DL uk (x) − DL u(x) gk (x) = − . 2 2 By convexity gk ≥ 0 and gk (x) → |DL u(x)|p for a.a. x ∈ Ω. Therefore, by the Fatou lemma and (3.3.15) we have p Z Z D u − D u L k L dx kDL ukpp ≤ lim inf gk dx = kDL ukpp − lim sup k→∞ 2 k→∞ Ω Ω 1 = kDL ukpp − p lim sup kDL uk − DL ukpp . 2 k→∞ Hence, lim supk→∞ kDL uk − DL ukpp = 0, that is limk→∞ kuk − uk = 0 by the definition of the norm k · k of the space W0L,p (Ω). 2 Remark 3.3.3. In the case d = 1 the conclusion of the previous theorem follows easily from Proposition 1.2.3. Indeed, since the space W0L,p (Ω) with the norm k · k is uniformly convex, by (3.3.15) and the fact that uk * u weakly in W0L,p (Ω) we get lim uk = u in W0L,p (Ω),

k→∞

thanks to Proposition 3.32 of [23]. The main result of the section is proved by using the energy functional Jλ associated to (3.1.1), which is given by Jλ (u) = Φ(u) + λΨ(u), where Ψ(u) = Ψ1 (u) + Ψ2 (u), Z 1 p Ψ1 (u) = − kukp,w , Ψ2 (u) = − F (x, u(x))dx. p Ω

(3.3.16)

42


It is easy to see that the functional Jλ is well defined in W0L,p (Ω) and of class C 1 (W0L,p (Ω)); see the proof of Lemma 3.3.2. Furthermore, Z 0 p hJλ (u), ϕi =M (kuk ) |DL u|p−2 DL uDL ϕdx Ω Z −λ w(x)|u(x)|p−2 u(x) + f (x, u(x)) ϕ(x)dx. Ω

Therefore, the critical points u ∈ W0L,p (Ω) of the functional Jλ are exactly the weak solutions of problem (3.1.1). Using the notation of Section 3.2, if Ψ(v) < 0 at some v ∈ W0L,p (Ω), then the crucial positive number λ? = ϕ1 (0) =

inf −1

u∈Ψ

− (I0 )

Φ(u) , Ψ(u)

I0 = (−∞, 0),

(3.3.17)

is well defined. Lemma 3.3.4. Ψ−1 (I0 ) is non-empty and λ? ≤ λ? < aλ1 . Proof. From (F)–(c) it follows that Ψ(u1 ) < −

M (λ1 ) < 0, apλ1

i.e. u1 ∈ Ψ−1 (I0 ),

(3.3.18)

since a > 0 by (3.1.2). Hence, λ? is well defined. By (3.3.18) λ? = ϕ1 (0) =

inf −1

u∈Ψ

− (I0 )

Φ(u) Φ(u1 ) M (λ1 )/p ≤ < = aλ1 , Ψ(u) −Ψ(u1 ) M (λ1 )/apλ1

as required. Finally, by (3.1.2), (3.3.2), (3.3.8) and (3.3.16), for all u ∈ W0L,p (Ω), with u 6≡ 0, we have M (kukp )/p a Φ(u) akukp ≥ ≥ λ1 = λ? , ≥ p p |Ψ(u)| (1 + Sf )kukp,w /p (1 + Sf )kukp,w 1 + Sf since a > 0 and b ≥ 0 by (3.1.2). Hence, in particular λ? ≥ λ? .

2

Lemma 3.3.5. The operators Ψ01 , Ψ02 , Ψ0 : W0L,p (Ω) → W ? are compact and Ψ1 , Ψ2 , Ψ are sequentially weakly continuous in W0L,p (Ω).

43

Proof. Of course, Ψ0 = Ψ01 + Ψ02 , where Z Z p−2 0 0 hΨ1 (u), vi = − w(x)|u| uv dx and hΨ2 (u), vi = − f (x, u)v dx, Ω

Ω

for all u, v ∈ W0L,p (Ω). Since Ψ01 and Ψ02 are continuous, thanks to the reflexivity of W0L,p (Ω) it is sufficient to show that Ψ01 and Ψ02 are weak-tostrong sequentially continuous, i.e. if (uk )k , u are in W0L,p (Ω) and uk * u in W0L,p (Ω), then kΨ01 (uk ) − Ψ01 (u)k? → 0 and kΨ02 (uk ) − Ψ02 (u)k? → 0 as k → ∞. To this aim, fix (uk )k ⊂ W0L,p (Ω), with uk * u in W0L,p (Ω). First, uk → u in Lp (Ω; w), since the embedding W0L,p (Ω) ,→ Lp (Ω; w) 0 is compact as shown above. Let Np : Lp (Ω; w) → Lp (Ω; w) denote the Nemytskii operator, defined by Np (u) = |u|p−2 u for all u ∈ Lp (Ω; w). 0 We claim that Np (uk ) → Np (u) in Lp (Ω; w) as k → ∞. Indeed, fix a subsequence (ukj )j of (uk )k . By Lemma 3.2.3, with A = Ω, N = n, s = p and ω = w, there exists a subsequence of (ukj )j , still denoted by (ukj )j , satisfying (i) and (ii). In particular, Np (ukj ) → Np (u) a.e. in Ω and 0 0 0 |Np (ukj )| ≤ hp−1 ∈ Lp (Ω; w), so that w|Np (ukj ) − Np (u)|p ≤ 2p whp ∈ L1 (Ω). Now, by the dominated convergence theorem, Np (ukj ) → Np (u) 0 in Lp (Ω; w). Therefore, the entire sequence (Np (uk ))k converges to Np (u) 0 in Lp (Ω; w) as k → ∞. Now, for all ϕ ∈ W0L,p (Ω), with kϕk = 1, we have, by Hölder’s inequality, Z 0 0 0 |hΨ1 (uk ) − Ψ1 (u), ϕi| ≤ w1/p |Np (uk ) − Np (u)| · w1/p |ϕ|dx Ω

≤ kNp (uk ) − Np (u)kp0,w kϕkp,w ≤ Sp,w kNp (uk ) − Np (u)kp0,w , where Sp,w is the Sobolev constant for the embedding W0L,p (Ω) ,→ Lp (Ω; w). Hence, kΨ01 (uk ) − Ψ01 (u)k? → 0 as k → ∞ and Ψ01 is compact. Let us prove now that also Ψ02 is compact. First, we show that the embedding W0L,p (Ω) ,→ Lq (Ω; w) is compact. Indeed, if n > Lp, then the 0 embedding W0L,p (Ω) ,→ L$ q (Ω) is compact, being $ > n/Lp, 1 < q < p 0 and so $0 q < p∗L . While, if n ≤ Lp, the embedding W0L,p (Ω) ,→ L$ q (Ω) is compact, being $ < ∞. In both cases, since w ∈ L$ (Ω), by Hölder’s 0 inequality, it is easy to show that L$ q (Ω) is continuously embedded in

44


Lq (Ω; w) and the claim is proved. Hence, uk → u in Lq (Ω; w). Define Nf : 0 Lq (Ω; w) → Lq (Ω; w1/(1−q) ) by Nf (u) = f (·, u(·)) for all u ∈ Lq (Ω; w). 0 We assert that Nf (uk ) → Nf (u) in Lq (Ω; w1/(1−q) ) as k → ∞. Indeed, fix a subsequence (ukj )j of (uk )k . There exists a subsequence, still denoted by (ukj )j , satisfying (i) and (ii) of Lemma 3.2.3, with A = Ω, N = n, s = p and ω = w, that is ukj → u a.e. in Ω and |ukj | ≤ h a.e. in Ω for all j ∈ N and some h ∈ Lq (Ω; w). In particular, |Nf (ukj ) − 0 Nf (u)|q w1/(1−q) → 0 a.e. in Ω, being f (x, ·) continuous for a.a. x ∈ 0 Ω. Furthermore, |Nf (ukj ) − Nf (u)|q w1/(1−q) ≤ κw(1 + hq ) ∈ L1 (Ω), 0 0 κ = (2Cf )q 2q −1 , by (F)–(a), being w ∈ L$ (Ω) ⊂ L1 (Ω), since $ > 1 and Ω is bounded. This shows the assertion, since 1 < q < p by (F)– (a). Hence, by the dominated convergence theorem, we have Nf (unk ) → 0 Nf (u) in Lq (Ω, w1/(1−q) ). Therefore the entire sequence Nf (uk ) → Nf (u) 0 in Lq (Ω, w1/(1−q) ) as k → ∞. Finally, for all ϕ ∈ W0L,p (Ω), with kϕk = 1, we have by Hölder’s inequality,

|hΨ02 (uk )

−

Ψ02 (u), ϕi|

Z ≤

w−1/q |Nf (uk ) − Nf (u)| · w1/q |ϕ|dx

Ω

≤ kNf (uk ) − Nf (u)kq0,w1/(1−q) kϕkq,w ≤ Sq,w kNf (uk ) − Nf (u)kq0,w1/(1−q) ,

where Sq,w is the Sobolev constant for the embedding W0L,p (Ω) ,→ Lq (Ω; w). Thus, kΨ02 (uk ) − Ψ02 (u)k? → 0 as k → ∞, that is Ψ02 is compact. Since by the above steps Ψ0 = Ψ01 + Ψ02 is compact, then Ψ is sequentially weakly continuous by Corollary 41.9 of [111], being W0L,p (Ω) reflexive. 2

Lemma 3.3.6. The functional Jλ (u) = Φ(u) + λΨ(u) is coercive for every λ ∈ (−∞, aλ1 ).

45

Proof. Fix λ ∈ (−∞, aλ1 ). Then by (3.1.2), (3.3.3) and (F)–(a) Z a λ p p Jλ (u) ≥ kuk − kukp,w − |λ| |F (x, u)|dx p p ZΩ λ 1 ≥ a− kukp − |λ| |F (x, u)|dx p λ1 Z ZΩ f1 (x) 1 λ p f0 (x) + ≥ a− kuk −|λ| f0 (x)dx −|λ| |u|q dx p λ1 q Ω Ω 1 λ ≥ a− kukp − |λ|C1 − |λ| C2 kukq , p λ1 q where C1 = kf0 k1 and C2 = S$ 0 q kf0 + f1 /qk$ , where S$ 0 q denotes the 0 Sobolev constant of the compact embedding W0L,p (Ω) ,→ L$ q (Ω). Note that C1 < ∞, since f0 ∈ L$ (Ω) ⊂ L1 (Ω), by (F)–(a), being $ > 1 and Ω bounded. This shows the assertion, since 1 < q < p by (F)–(a). 2

Theorem 3.3.7. The following statements hold. (i) If (F)–(a), (b) hold and λ ∈ [0, λ? ), where λ? is defined in (3.3.7), then (3.1.1) has only the trivial solution. (ii) If (F) holds from (a) to (c), then problem (3.1.1) admits at least two nontrivial solutions for every λ ∈ (λ? , a λ1 ), where λ? = ϕ1 (0) < a λ1 . Proof. (i) Let u ∈ W0L,p (Ω) be a nontrivial weak solution of the problem (3.1.1), then Z Z p−2 p λ1 M (kuk ) |DL u| DL uDL ϕ dx = λ1 λ {w(x)|u|p−2 u + f (x, u)}ϕ dx Ω

Ω

for all ϕ ∈ W0L,p (Ω). Take ϕ = u and by (3.1.2) and (3.3.6) Z p p p aλ1 kuk ≤ λ1 M (kuk )kuk = λ1 λ {w(x)|u|p + f (x, u)u}dx Ω Z f (x, u) p p = λ1 λ kukp,w + w(x)|u| dx p−1 Ω w(x)|u| ≤ λ1 λ(1 + Sf )kukpp,w ≤ λ(1 + Sf )kukp

46


by (3.3.3). Therefore λ ≥ λ? , as required. (ii) The functional Φ is convex. Moreover, as we already proved in Lemma 3.3.2, Φ is weakly lower semicontinuous in W0L,p (Ω) and Φ0 verifies condition (S+ ). Furthermore, Ψ0 : W0L,p (Ω) → W ? is compact and Ψ is sequentially weakly continuous in W0L,p (Ω) by Lemma 3.3.5. The functional Jλ is coercive for every λ ∈ I, where I = (−∞, aλ1 ), thanks to Lemma 3.3.6. We claim that Ψ(W0L,p (Ω)) ⊃ R− 0 . Indeed, Ψ(0) = 0 and by (F)–(a) Z Z |u(x)| Z Z Z |u(x)| f0 + f1 |s|q−1 dsdx |f (x, s)|dsdx ≤ |F (x, u)|dx ≤ Ω Ω 0 ZΩ 0 f1 f0 |u(x)| + |u(x)|q dx. = q Ω Divide the domain Ω into two parts Ω1 = {x ∈ Ω : |u(x)| ≤ 1}

Ω2 = {x ∈ Ω : |u(x)| ≥ 1}

and calculate Z f1 q f0 |u(x)| + |u(x)| dx q Ω Z Z f1 q f1 q q ≤ f0 + |u| dx + f0 |u| + |u| dx q q Ω1 Ω2 Z Z Cf 1 q w(x)|u| dx + w(x)|u|q dx ≤ kf0 k1 + Cf 1 + q Ω1 q Ω2 Z Z Cf q w(x)|u| dx + Cf w(x)|u|q dx = kf0 k1 + q Ω Z ZΩ2 Cf ≤ kf0 k1 + w(x)|u|q dx + Cf w(x)|u|q dx q Ω Ω Z q ≤ kf0 k1 + 2Cf w(x)|u| dx. Ω

Furthermore, by Hölder’s inequality Z Z (p−q)/p q w(x)|u| dx = w(x)q/p |u|q w(x)1−q/p dx ≤ kwk1 kukqp,w , Ω

Ω

47

since w ∈ L1 (Ω), being $ > 1 and Ω bounded. Hence, combining together the previous estimates, we get Z 1 p Ψ(u) ≤ − kukp,w + |F (x, u)|dx p Ω Z 1 p w(x)|u|q dx ≤ − kukp,w + kf0 k1 + 2Cf p Ω 1 (p−q)/p ≤ − kukpp,w + kf0 k1 + 2Cf kwk1 kukqp,w . p Therefore, lim

u∈W0L,p (Ω) kukp,w →∞

Ψ(u) = −∞,

since q < p. Hence, the claim follows by the continuity of Ψ. −1 −1 Thus, (inf Ψ, sup Ψ) ⊃ R− 0 and so Ψ (I0 ) 6= ∅. For every u ∈ Ψ (I0 ) we have ϕ1 (r) ≤

inf v∈Ψ−1 (r) Φ(v) − Φ(u) Ψ(u) − r

so that lim sup ϕ1 (r) ≤ − r→0−

Φ(u) Ψ(u)

for all r ∈ (Ψ(u), 0),

for all u ∈ Ψ−1 (I0 ),

in other words, lim sup ϕ1 (r) ≤ ϕ1 (0) = λ? .

(3.3.19)

r→0−

From (F)–(a) and (b), that is (3.3.5) and (3.3.6), it follows the existence of a positive real number L > 0 such that ?

|F (x, s)| ≤ Lw(x)|s|p

for a.a. x ∈ Ω and all s ∈ R.

(3.3.20)

To this aim, denoting by `0 the limit number in (3.3.5), there exists ? δ > 0 such that |F (x, s)| ≤ (`0 + 1)w(x)|s|p for a.a. x ∈ Ω and all s, with |s| < δ. Fix s, with |s| ≥ δ, then by (3.3.8) for a.a. x ∈ Ω ?

Sf p−p? Sf δ p−p ? ? |F (x, s)| ≤ |s| w(x)|s|p ≤ w(x)|s|p , p p being p? ∈ (p, p∗L /$0 ). ? 1, Sf δ p−p /p}.

Hence, (3.3.20) holds, with L = max{`0 +

48

Chapter 3. Eigenvalue Systems ?

We note in passing that the embedding W0L,p (Ω) ,→ Lp (Ω; w) is continuous. Indeed, by Hölder’s inequality Z ? ? p? ˜ w(x)|u|p dx ≤ |Ω|1/℘ kwk$ kukpp∗ ≤ Ckuk , (3.3.21) L

Ω

? where C˜ = Spp∗ |Ω|1/℘ kwk$ , Sp∗L is the Sobolev constant for the embedding L

∗

W0L,p (Ω) ,→ LpL (Ω) and ℘ is the crucial exponent $0 p∗L , if n > Lp, ℘ = p∗L − p? $0  0 $, if 1 ≤ n ≤ Lp.  

Note that ℘ > 1, being p? ∈ (p, p∗L /$0 ) by (F)–(b). Hence, by (3.3.20) and (3.3.21) for every u ∈ W0L,p (Ω), we get |Ψ(u)| ≤

1 ? kukp + Cp? kukp , pλ1

(3.3.22)

where Cp? = C˜ L. Therefore, given r < 0 and v ∈ Ψ−1 (r), condition (3.1.2) yields a r = a Ψ(v) ≥ −

a 1 ? ? kvkp −aCp? kvkp ≥ − Φ(v)−κΦ(v)p /p , (3.3.23) pλ1 λ1

?

?

where κ = a1−p /p Cp? pp /p . Since the functional Φ is bounded below, coercive and lower semicontinuous on the reflexive Banach space W0L,p (Ω), it is easy to see that Φ is also coercive on the sequentially weakly closed non-empty set Ψ−1 (r), see Lemma 3.3.5. Therefore, by Theorem 6.1.1 of [18], there exists an element ur ∈ Ψ−1 (r) such that Φ(ur ) = inf Φ(v). −1 v∈Ψ

(r)

By (3.2.2), we have ϕ2 (r) ≥ −

1 Φ(ur ) inf Φ(v) = , −1 r v∈Ψ (r) |r|

49

being u ≡ 0 ∈ Ψ−1 (I r ). From (3.3.23) we get p? /p Φ(ur ) 1 Φ(ur ) p? /p−1 a≤ · + κ|r| λ1 |r| |r| ϕ2 (r) ? ? + κ|r|p /p−1 ϕ2 (r)p /p . ≤ λ1

(3.3.24)

There are now two possibilities to be considered: either ϕ2 is locally bounded at 0− , so that the above inequality shows at once that lim inf ϕ2 (r) ≥ aλ1 , − r→0

being p? > p, or lim supr→0− ϕ2 (r) = ∞. In both cases (3.3.19) and Lemma 3.3.4 yield that for all integers k ≥ k ? = 1 + [2/(aλ1 − λ? )] there exists a number rk < 0 so close to zero that ϕ1 (rk ) < λ? + 1/k < aλ1 − 1/k < ϕ2 (rk ). Hence, by Theorem 3.2.1, Part (a) of (ii), being u ≡ 0 a critical point of Jλ , problem (3.1.1) admits at least two nontrivial solutions for all λ ∈ (λ? , aλ1 ) =

∞ [

∞ [

[λ? + 1/k, aλ1 − 1/k] ⊂

k=k?

(ϕ1 (rk ), ϕ2 (rk )) ∩ I,

k=k?

2

as claimed. In this final part of the section, we consider the problem ( M (kukp )∆Lp u = λf (u) in Ω, u=0 on ∂Ω,

(3.3.25)

with M satisfying (3.1.2) and f verifying conditions (F)–(a), (b) and instead of condition (F)–(c), we require the less stringent assumption (F)–(c0 ) Assume that there exist x0 ∈ Ω, s0 ∈ Rd and r0 > 0 so small that B0 = {x ∈ Rn : |x − x0 | ≤ r0 } ⊂ Ω, ess inf F (x, s0 ) = µ0 > 0, B0

ess sup max |F (x, s)| = M0 < ∞. B0

|s|≤|s0 |

Clearly, when f does not depend on x, condition (F)–(c0 ) simply reduces to the request that F (s0 ) > 0 at a point s0 ∈ Rd .

50


Corollary 3.3.8. Suppose that f : Ω × R → R satisfies (F)–(a), (b). (i) If λ ∈ [0, `? ), where `? = aλ1 /Sf , then (3.3.25) has only the trivial solution. (ii) If furthermore f satisfies (F)–(c0 ), then there exists `? ≥ `? such that (3.3.25) admits at least two nontrivial solutions for all λ ∈ (`? , ∞). Proof. The energy functional Jλ associated to (3.3.25), is given by Jλ (u) = Φ(u) + λΨ2 (u), where Z Ψ2 (u) = − F (x, u(x))dx Ω

and Φ(u) is defined in the statement of Lemma 3.3.2. First, note that Jλ is coercive for every λ ∈ R. Indeed, as in the proof of Lemma 3.3.6 Z a a p Jλ (u) ≥ kuk − |λ| |F (x, u)|dx ≥ kukp − |λ|C1 − |λ| C2 kukq , p p Ω q where C1 = kf0 k1 , C2 = S$ 0 q kf0 + f1 /qk$ and S$ 0 q denotes the Sobolev 0 constant of the compact embedding W0L,p (Ω) ,→ L$ q (Ω). This shows the claim, since 1 < q < p by (F)–(a). Hence, here I = R. The part (i) of the statement is proved by using the same argument produced for the proof of Theorem 3.3.7–(i), being Z p p p aλ1 kuk ≤ λ1 M (kuk )kuk dx = λ1 λ f (x, u)udx Ω

≤ λ1 λSf kukpp,w ≤ λSf kukp by (3.3.2). Thus, if u is a nontrivial weak solution of (3.3.25), then necessarily λ ≥ `? = aλ1 /Sf , as required. In order to prove (ii), we first show that there exists u0 ∈ W0L,p (Ω) such that Ψ2 (u0 ) < 0, so that the crucial number `? = ϕ1 (0) =

inf −1

u∈Ψ2 (I0 )

−

Φ(u) , Ψ2 (u)

I0 = (−∞, 0),

(3.3.26)

51

is well defined. Note that s0 6= 0 in (F)–(c0 ). Take σ ∈ (0, 1) and put B = {x ∈ Rn : |x − x0 | ≤ σr0 }. Of course, B ⊂ B0 . Consider a function u0 ∈ [Cc∞ (Ω)]d such that |u0 | ≤ |s0 | in Ω,

supp u0 ⊂ B0

and u0 ≡ s0 in B.

Clearly, u0 ∈ W0L,p (Ω). Now, by (F)–(c0 ), Z Z Ψ2 (u0 ) = − F (x, u0 (x))dx − F (x, s0 )dx ≤ M0 |B0 \ B| − µ0 |B| =

B0 \B ωn r0n [M0 (1

B n

n

− σ ) − µ0 σ ] ,

where ωn is the measure of the unit ball in Rn . Hence, taking σ ∈ (0, 1) so large that σ n > M0 /(µ0 + M0 ), we get that Ψ2 (u0 ) < 0, as claimed. Furthermore, by (3.1.2), (3.3.3) and (3.3.8), for all u ∈ W0L,p (Ω), with u 6≡ 0, we have Φ(u) akukp a ≥ ≥ λ1 = `? . p |Ψ2 (u)| Sf kukp,w Sf Therefore, by (3.3.26), `? ≥ `? . In particular, for all u ∈ Ψ−1 2 (I0 ), we have ϕ1 (r) ≤

inf v∈Ψ−1 Φ(v) − Φ(u) 2 (r) Ψ2 (u) − r

for all r ∈ (Ψ2 (u), 0).

Hence, lim sup ϕ1 (r) ≤ ϕ1 (0) = `? . r→0−

Also in this setting (3.3.20) and (3.3.21) hold and (3.3.22) simply reduces to ? |Ψ2 (u)| ≤ Cp? kukp . Taken r < 0 and v ∈ Ψ−1 2 (r), we obtain ?

r = Ψ2 (v) ≥ −Cp? kvkp ≥ −Cp?

p a

p? /p Φ(v) .

r Therefore, by (3.2.2), since u ≡ 0 ∈ Ψ−1 2 (I ),

ϕ2 (r) ≥

1 |r|

inf −1

v∈Ψ2 (r)

? −1

Φ(v) ≥ κ|r|p/p

,

52

Chapter 3. Eigenvalue Systems −p/p?

where κ = a Cp?

/p. This implies that lim− ϕ2 (r) = ∞, being p? > p. r→0

In conclusion, we have proved that lim sup ϕ1 (r) ≤ ϕ1 (0) = `? < lim− ϕ2 (r) = ∞.

(3.3.27)

r→0

r→0−

This shows that for all integers k ≥ k ? = 2 + [`? ] there exists rk < 0 so close to zero that ϕ1 (rk ) < `? + 1/k < k < ϕ2 (rk ). Hence, since I = R, by Theorem 3.2.1, Part (a) of (ii), being u ≡ 0 a critical point of Jλ , problem (3.3.25) admits at least two nontrivial solutions for all λ∈

∞ [

(ϕ1 (rk ), ϕ2 (rk )) ⊃

k=k?

∞ [

[`? + 1/k, k] = (`? , ∞),

k=k?

2

as claimed.

It is apparent from the main definitions (3.3.7), (3.3.17), Corollary 3.3.8 and (3.3.26) that 0 < λ? < `? ≤ `? ≤ λ? . Hence, Corollary 3.3.8 provides also the useful information that 0 < λ? < λ? .

3.4

Complements to Chapter 3

In this section we study the eigenvalue problems (3.1.4) and (3.1.5), under Dirichlet and Robin boundary conditions respectively. For simplicity, we consider only the case of the equations, that is here d = 1. These two problems are governed by the same operator A. Let A : Ω × Rn → R be the potential of A, satisfying the next assumption. (A) Let A : Ω × Rn → R, A = A (x, ξ), be a continuous function in Ω × Rn , with continuous derivative with respect to ξ, A = ∂ξ A , and suppose that the following conditions hold. (a) A (x, 0) = 0 and A (x, ξ) = A (x, −ξ) for all x ∈ Ω and ξ ∈ Rn . (b) A (x, ·) is strictly convex in Rn for all x ∈ Ω. (c) There exist two constants c, C, with 0 < c ≤ C, such that A(x, ξ) · ξ ≥ c|ξ|p for all (x, ξ) ∈ Ω × Rn .

and

|A(x, ξ)| ≤ C|ξ|p−1

53

Remark 3.4.1. Of course A(x, ξ) · ξ ≥ A (x, ξ) for all (x, ξ) ∈ Ω × Rn when A verifies (A)–(a), (b). Moreover, by (A)–(a) and (c) c|ξ|p ≤ pA (x, ξ) ≤ C|ξ|p

(3.4.1)

R1 for all (x, ξ) ∈ Ω×Rn . Indeed, by (A)–(a) it results A (x, ξ) = 0 A(x, tξ)· ξdt. Hence, (A)–(c) implies Z 1 Z 1 1 c p p−1 p t dt ≤ |ξ| = c|ξ| A(x, tξ) · tξdt = A (x, ξ) p 0 t 0 Z 1 Z 1 C p tp−1 dt = |ξ|p , |A(x, tξ)|dt ≤ C|ξ| ≤ |ξ| p 0 0 and (3.4.1) is proved. The potential A (x, ξ) = |ξ|p /p of A(x, ξ) = |ξ|p−2 ξ satisfies (A) for all p > 1. The corresponding divergence operator is the usual p-Laplacian operator ∆p u = div(|Du|p−2 Du), so that ∆p can be covered in this paper for all p > 1. Condition (b) is weaker than the request that A is p-uniformly convex, i.e. that there exists a constant k > 0 such that ξ+η 1 1 0 (b ) A x, ≤ A (x, ξ) + A (x, η) − k|ξ − η|p 2 2 2 for all x ∈ Ω and ξ, η ∈ Rn . This is indeed the standard condition assumed in this context in the literature and which forces that p ≥ 2 when A (x, ξ) = |ξ|p /p, cf. [43]. Taking inspiration from the famous Lemma 3 of [42], we prove the crucial result. Lemma 3.4.2. Let A verify (A) from (a) to (c) and let ξ, (ξk )k be in Rn such that (A(x, ξk ) − A(x, ξ)) · (ξk − ξ) → 0 as k → ∞.

(3.4.2)

Then (ξk )k converges to ξ. Proof. First we assert that (ξk )k is bounded. Otherwise, up to a subsequence, still denoted by (ξk )k , we would have |ξk | → ∞. Hence, by (A)–(c) (A(x, ξk ) − A(x, ξ)) · (ξk − ξ) → ∞,

54


which is impossible by (3.4.2). Therefore, (ξk )k is bounded and possesses a subsequence, still denoted by (ξk )k , which converges to some η ∈ Rn . Thus (A(x, η) − A(x, ξ)) · (η − ξ) = 0 by (3.4.2) and the strict convexity of A (x, ·) for all x ∈ Ω implies that η = ξ. This also shows that actually the entire sequence (ξk )k converges to ξ. 2 Existence of Three Solutions for Dirichlet Eigenvalue Problems Involving a Divergence Type Operator In this subsection, we study problem (3.1.4), so that Ω is a bounded domain of Rn , n ≥ 1, p > 1, X = W01,p (Ω), k · k = kD · kp . In this setting, Z Φ(u) = A (x, Du(x))dx, Φ : W01,p (Ω) → R. (3.4.3) Ω

Adopting an argument given in the proof of Lemma 4.2 of [32], we prove the following useful result. Lemma 3.4.3. Let A satisfy (A) from (a) to (c). Then the functional Z Φ(u) = A (x, Du(x))dx, Φ : W01,p (Ω) → R, Ω

is convex, weakly lower semicontinuous and of class C 1 in W01,p (Ω). 0 Moreover, Φ0 : W01,p (Ω) → W −1,p (Ω) verifies the (S+ ) condition, i.e. 1,p for every sequence (uk )k ⊂ W0 (Ω) such that uk * u weakly in W01,p (Ω) and Z (3.4.4) lim sup A(x, Duk ) · (Duk − Du)dx ≤ 0, k→∞

Ω

then uk → u strongly in W01,p (Ω). Proof. A simple calculation shows that the functional Φ is convex and of class C 1 in W01,p (Ω) (the proof is similar to the one given in Lemma 3.3.2). Hence, in particular Φ is weakly lower semicontinuous in W01,p (Ω), see Corollary 3.9 of [23]. Let (uk )k be a sequence in W01,p (Ω) as in the statement. Then Φ(u) ≤ lim inf Φ(uRk ), being Φ weakly lower semicontinuous on W01,p (Ω). Furthermore, Ω A(x, Du) · (Duk − Du)dx → 0 as k → ∞, since uk * u in

55

W01,p (Ω) as k → ∞, so that Duk * Du in [Lp (Ω)]n , and |A(x, Du)| ∈ 0 Lp (Ω) by (A)–(c). R Therefore, by convexity and (3.4.4) we get that 0 ≤ lim supk→∞ Ω (A(x, Duk ) − A(x, Du)) · (Duk − Du)dx ≤ 0. In other words Z lim (A(x, Duk ) − A(x, Du)) · (Duk − Du)dx = 0, k→∞

Ω

that is the sequence (A(x, Duk ) − A(x, Du)) · (Duk − Du) ≥ 0 converges to 0 in L1 (Ω). Hence, up to a subsequence, (A(x, Duk ) − A(x, Du)) · (Duk − Du) → 0 a.e. in Ω and by Lemma 3.4.2 also Duk → Du a.e. in Ω. In particular, (3.4.4) holds in the stronger form Z lim A(x, Duk ) · (Duk − Du)dx = 0. (3.4.5) k→∞

Ω

By the convexity of Φ for all k Z Φ(u) + A(x, Duk ) · (Duk − Du)dx ≥ Φ(uk ), Ω

so that Φ(u) ≥ lim supk→∞ Φ(uk ) by (3.4.5). In conclusion, Φ(u) = lim Φ(uk ). k→∞

(3.4.6)

Consider the sequence (gk )k in L1 (Ω) defined pointwise by gk (x) = 21 [A (x, Duk (x)) + A (x, Du(x))] − A x, 12 [Duk (x) − Du(x)] . By (A)–(a) and convexity gk ≥ 0 and gk (x) → A (x, Du(x)) for a.a. x ∈ Ω, being A (x, ·) continuous for a.a. x ∈ Ω. Therefore, by the Fatou lemma and (3.4.6) we have Z Z Φ(u) ≤ lim inf gk (x)dx = Φ(u)−lim sup A x, 12 [Duk (x) − Du(x)] dx. k→∞

k→∞

Ω

Ω

1 A x, [Du (x) − Du(x)] dx ≤ 0, that is k 2 Ω Z lim A x, 12 [Duk (x) − Du(x)] dx = 0,

Hence, lim supk→∞

R

k→∞

Ω

56


in other words limk→∞ kDuk − Dukp = 0 by (3.4.1). In conclusion, lim kuk − uk = 0,

k→∞

2

as required.

Before introducing the main structural assumptions on f , let us recall some basic properties. As in Section 3.3 it is easy to show that also the embedding W01,p (Ω) ,→ Lp (Ω; w) is compact. Let λ1 be the first eigenvalue of the problem −∆p u = λ w(x)|u|p−2 u in W01,p (Ω), that is λ1 is defined by the Rayleigh quotient R |Du|p dx Ω R λ1 = inf . p u∈W01,p (Ω) Ω w(x)|u| dx

(3.4.7)

u6=0

By Proposition 3.1 of [37], the infimum in (3.4.7) is achieved and λ1 > 0. In particular, λ1 kukpp,w ≤ kukp

for every u ∈ W01,p (Ω).

(3.4.8)

Denote by u1 ∈ W01,p (Ω) the normalized eigenfunction corresponding to λ1 , that is ku1 kp,w = 1. Also in this setting on f we assume (F)–(a) and (b), with the only difference that in this case clearly p? ∈ (p, p∗ /$0 ), being L = 1 Moreover, in place of (F)–(c) we require the next condition. Z Z s 1 C F (x, u1 (x))dx > f (x, t)dt (F)–(c1 ) − 1 , where F (x, s) = p c Ω 0 and u1 is the first normalized eigenfunction defined above, c, C > 0 are the constants given in (A)–(c). Also here condition (F) implies that f (x, 0) = 0 for a.a. x ∈ Ω and that (3.3.5), (3.3.6) and (3.3.8) hold. Hence the positive number λ? =

c λ1 , 1 + Sf

(3.4.9)

57

where c > 0 is the smaller constant introduced in (A)–(c) and Sf is given in (3.3.6), is well defined. The main result of the section is proved by using the energy functional Jλ associated to (3.1.4), which is given by Jλ (u) = Φ(u)+λΨ(u), where Φ is defined in (3.4.3) and Ψ in (3.3.16). It is easy to see that the functional Jλ is well defined in W01,p (Ω) and of class C 1 . Furthermore, Z 0 hJλ (u), ϕi = A(x, Du(x))Dϕ(x)dx Ω Z w(x)|u(x)|p−2 u(x) + f (x, u(x)) ϕ(x)dx, −λ Ω

where, h·, ·i denotes the duality pairing between W01,p (Ω) and its dual 0 space W −1,p (Ω). Therefore, the critical points u ∈ W01,p (Ω) of the functional Jλ are the weak solutions of problem (3.1.4). Using the notation of Section 3.2, if Ψ(v) < 0 at some v ∈ W01,p (Ω), then the crucial positive number λ? = ϕ1 (0) =

inf −1

u∈Ψ

− (I0 )

Φ(u) , Ψ(u)

I0 = (−∞, 0),

(3.4.10)

with the new definition of Φ, is well defined. Lemma 3.4.4. Ψ−1 (I0 ) is non-empty and λ? ≤ λ? < cλ1 . Proof. From (F)–(c1 ) it follows that 1 1 C C − 1 = − < 0, Ψ(u1 ) < − − p p c cp

i.e. u1 ∈ Ψ−1 (I0 ),

being C ≥ c > 0 by (3.4.1). Hence, λ? is well defined. Again by (A)–(c) and (F)–(c) R p Ω A (x, Du1 )dx Φ(u) Φ(u1 ) ? λ = ϕ1 (0) = inf − ≤− = u∈Ψ−1 (I0 ) Ψ(u) Ψ(u1 ) −pΨ(u1 ) p Cku1 k < = cλ1 , C/c as required. Finally, by (3.4.1), (3.3.8) and (3.4.7), for all u ∈ W01,p (Ω), with u 6= 0, we have Φ(u) ckukp c ≥ ≥ λ1 = λ? . p |Ψ(u)| (1 + Sf )kukp,w 1 + Sf

58


2

Hence, in particular λ? ≥ λ? . 0

Lemma 3.4.5. The operators Ψ01 , Ψ02 , Ψ0 : W01,p (Ω) → W −1,p (Ω) are compact and Ψ1 , Ψ2 , Ψ are sequentially weakly continuous in W01,p (Ω). The proof is analogous to the argument produced for Lemma 3.3.5, with the only difference that here L = 1, and we omit it. Lemma 3.4.6. The functional Jλ (u) = Φ(u) + λΨ(u) is coercive for every λ ∈ (−∞, cλ1 ). Proof. Fix λ ∈ (−∞, cλ1 ). Then by (3.4.1), (3.4.8) and (F)–(a) Z λ c p p Jλ (u) ≥ kuk − kukp,w − |λ| |F (x, u)|dx p p ZΩ 1 λ ≥ c− kukp − |λ| |F (x, u)|dx p λ1 ZΩ Z λ f1 (x) 1 p c− kuk − |λ| f0 (x)dx − |λ| f0 (x) + |u|q dx ≥ p λ1 q Ω Ω 1 λ ≥ c− kukp − |λ|C1 − |λ| C2 kukq , p λ1 q where C1 = kf0 k1 and C2 = S$ 0 q kf0 + f1 /qk$ , where S$ 0 q denotes the 0 Sobolev constant of the compact embedding W01,p (Ω) ,→ L$ q (Ω). Note that C1 < ∞, since f0 ∈ L$ (Ω) ⊂ L1 (Ω), by (F)–(a), being $ > 1 and Ω bounded. This shows the assertion, since 1 < q < p by (F)–(a). 2

Theorem 3.4.7. Assume (A) from (a) to (c). (i) If (F)–(a), (b) hold and λ ∈ [0, λ? ), where λ? is defined in (3.4.9), then (3.1.4) has only the trivial solution. (ii) If (F)–(a), (b) and (c1 ) hold, then problem (3.1.4) admits at least two nontrivial solutions for every λ ∈ (λ? , c λ1 ), where λ? = ϕ1 (0) < c λ1 . Proof. (i) Let u ∈ W01,p (Ω) be a nontrivial weak solution of the problem (3.1.4), then Z Z A(x, Du)Dϕ dx = λ {w(x)|u|p−2 u + f (x, u)}ϕ dx Ω

Ω

59

for all ϕ ∈ W01,p (Ω). Take ϕ = u and by (3.4.1) and (3.3.6) Z Z p cλ1 kuk ≤ λ1 A(x, Du)Du dx = λ1 λ {w(x)|u|p + f (x, u)u}dx Ω Ω Z f (x, u) p p = λ1 λ kukp,w + w(x)|u| dx p−1 Ω w(x)|u| ≤ λ1 λ (1 + Sf ) kukpp,w ≤ λ (1 + Sf ) kukp by (3.4.7). Therefore λ ≥ λ? , as required. (ii) By (A)–(b), the functional Φ is convex, moreover, by Lemma 3.4.3, Φ is weakly lower semicontinuous in W01,p (Ω) and Φ0 verifies condition 0 (S+ ). Furthermore, Ψ0 : W01,p (Ω) → W −1,p (Ω) is compact and Ψ is sequentially weakly continuous in W01,p (Ω) by Lemma 3.4.5. The functional Jλ is coercive for every λ ∈ I, where I = (−∞, cλ1 ), thanks to Lemma 3.4.6. Also in this setting it is possible to prove, as in the proof of Theorem 3.3.7, that (3.3.19) and (3.3.20) hold, namely lim sup ϕ1 (r) ≤ ϕ1 (0) = λ? , r→0−

|F (x, s)| ≤ Lw(x)|s|γ

for a.a. x ∈ Ω and all s ∈ R.

Therefore, (3.3.21) holds with L = 1, that is Z ? ? p? ˜ w(x)|u|p dx ≤ |Ω|1/℘ kwk$ kukpp∗ ≤ Ckuk , Ω ?

where C˜ = Spp∗ |Ω|1/℘ kwk$ , Sp∗ is the Sobolev constant for the embedding ∗ W01,p (Ω) ,→ Lp (Ω) and ℘ is the crucial exponent  0 ∗  $p , if n > p, ℘ = p∗ − p? $0  0 $, if n ≤ p. Note that ℘ > 1, being p? ∈ (p, p∗ /$0 ) by (F)–(b). Hence, for every u ∈ W01,p (Ω), we get (3.3.22), that is |Ψ(u)| ≤

1 ? kukp + Cp? kukp . pλ1

60


Therefore, given r < 0 and v ∈ Ψ−1 (r) condition (3.4.1) yields c r = c Ψ(v) ≥ −

1 c ? ? kvkp −cCp? kvkp ≥ − Φ(v)−κΦ(v)p /p , (3.4.11) pλ1 λ1

?

?

where κ = c1−p /p Cp? pp /p . Since the functional Φ is bounded below, coercive and lower semicontinuous on the reflexive Banach space W01,p (Ω), it is also coercive on the sequentially weakly closed non-empty set Ψ−1 (r) if the set Ψ−1 (r) is unbounded, otherwise no further assumptions on Φ are required; see Lemma 3.4.5. Therefore, by Theorem 6.1.1 of [18], there exists an element ur ∈ Ψ−1 (r) such that Φ(ur ) = inf Φ(v). By −1 v∈Ψ

(r)

(3.2.2), we have ϕ2 (r) ≥ −

1 Φ(ur ) inf Φ(v) = , −1 r v∈Ψ (r) |r|

being u ≡ 0 ∈ Ψ−1 (I r ). From (3.4.11) we get p? /p Φ(ur ) 1 Φ(ur ) p? /p−1 · + κ|r| c≤ λ1 |r| |r| ϕ2 (r) ? ? ≤ + κ|r|p /p−1 ϕ2 (r)p /p . λ1

(3.4.12)

There are now two possibilities to be considered: either ϕ2 is locally bounded at 0− , so that the above inequality shows at once that lim inf ϕ2 (r) ≥ cλ1 , − r→0

being p? > p, or lim supr→0− ϕ2 (r) = ∞. In both cases we have, as in the proof of Theorem 3.3.7, that for all integers k ≥ k ? = 1 + [2/(cλ1 − λ? )] there exists a number rk < 0 so close to zero that ϕ1 (rk ) < λ? + 1/k < cλ1 − 1/k < ϕ2 (rk ). Hence, by Theorem 3.2.1, Part (a) of (ii), being u ≡ 0 a critical point of Jλ , problem (3.1.4) admits at least two nontrivial solutions for all ?

λ ∈ (λ , cλ1 ) =

∞ [ k=k?

as claimed.

?

[λ + 1/k, cλ1 − 1/k] ⊂

∞ [

(ϕ1 (rk ), ϕ2 (rk )) ∩ I,

k=k?

2

61

Remark 3.4.8. In [70] the authors consider the following problem ( −divA(x, Du) = λf (u) in Ω, u=0 on ∂Ω, under the assumptions (A) from (a), (b) and (c0 ), given in Remark 3.4.1, the requirement that f : R → R is a continuous function satisfying (F)– (a), (b) in a form in which in particular f0 = f1 = constant, while (F)–(c1 ) is replaced by the condition that F (s0 ) > 0 at some point s0 ∈ R. In a similar way, instead of condition (F)–(c1 ), we consider the less stringent assumption (F)–(c01 ) Assume there exist x0 ∈ Ω, s0 ∈ R and r0 > 0 so small that B0 = {x ∈ Rn : |x − x0 | ≤ r0 } ⊂ Ω, ess inf F (x, |s0 |) = µ0 > 0, B0

ess sup max |F (x, t)| = M0 < ∞. B0

|t|≤|s0 |

Clearly, when f does not depend on x, condition (F)–(c01 ) simply reduces to the request that F (s0 ) > 0 at a point s0 ∈ R, as assumed in [70]. In this new setting, we have the following corollary which improves the main result of [70]. Corollary 3.4.9. Assume (A) from (a) to (c) and suppose that f : Ω × R → R satisfies (F)–(a), (b). Consider the problem ( −divA(x, Du) = λf (x, u) in Ω, (3.4.13) u=0 on ∂Ω, (i) If λ ∈ [0, `? ), where `? = cλ1 /Sf , then (3.4.13) has only the trivial solution. (ii) If furthermore f satisfies (F)–(c01 ), then there exists `? ≥ `? such that (3.4.13) admits at least two nontrivial solutions for all λ ∈ (`? , ∞). Proof. The energy functional Jλ associated to (3.4.13), is given by Jλ (u) = Φ(u) + λΨ2 (u), where Z Ψ2 (u) = − F (x, u(x))dx. Ω

62


First, note that Jλ is coercive for every λ ∈ R. Indeed, as shown in the proof of Lemma 3.4.6 Z c c p Jλ (u) ≥ kuk − |λ| |F (x, u)|dx ≥ kukp − |λ|C1 − |λ| C2 kukq , p p Ω q where C1 = kf0 k1 , C2 = S$ 0 q kf0 + f1 /qk$ and S$ 0 q denotes the Sobolev 0 constant of the compact embedding W01,p (Ω) ,→ L$ q (Ω). This shows the claim, since 1 < q < p by (F)–(a). Hence, here I = R. The part (i) of the statement is proved using the same argument produced for the proof of Theorem 3.4.7–(i), being Z Z p cλ1 kuk ≤ λ1 A(x, Du)Dudx = λ1 λ f (x, u)udx Ω

≤

Ω

λ1 λSf kukpp,w

≤ λSf kuk

p

by (3.4.7). Thus, if u is a nontrivial weak solution of (3.4.13), then necessarily λ ≥ `? = cλ1 /Sf , as required. In order to prove (ii), we first show that there exists u0 ∈ W01,p (Ω) such that Ψ2 (u0 ) < 0, so that the crucial number `? = ϕ1 (0) =

−

inf −1

u∈Ψ2 (I0 )

Φ(u) , Ψ2 (u)

I0 = (−∞, 0),

(3.4.14)

is well defined. Next, note that s0 6= 0 in (F)–(c01 ). Then take σ ∈ (0, 1) and put B = {x ∈ Rn : |x − x0 | ≤ σr0 }. Of course, B ⊂ B0 . Define  if x ∈ Ω \ B0 ,  0,  if x ∈ B, u0 (x) = |s0 |,  |s | 0   (r0 − |x − x0 |) , if x ∈ B0 \ B. r0 (1 − σ) Clearly, u0 ∈ C0 (Ω), 0 ≤ u0 (x) ≤ |s0 | for all x ∈ Ω and u0 ∈ W01,p (Ω). Furthermore, Z 1 − σn p ku0 k = |Du0 (x)|p dx = ωn r0n−p |s0 |p , (1 − σ)p Ω where ωn is the measure of the unit ball in Rn . Now, by (F)–(c01 ), Z Z Ψ2 (u0 ) ≤ M0 dx − F (x, |s0 |)dx ≤ ωn r0n [M0 (1 − σ n ) − µ0 σ n ] . B0 \B

B

63

Hence, taking σ ∈ (0, 1) so large that σ n > M0 /(µ0 + M0 ), we get that Ψ2 (u0 ) < 0, as claimed. Furthermore, by (3.4.1), (3.3.16), (3.3.8) and (3.4.8), for all u ∈ 1,p W0 (Ω), with u 6≡ 0, we have Φ(u) ckukp c ≥ ≥ λ1 = `? . p |Ψ2 (u)| Sf kukp,w Sf Hence, `? ≥ `? . In particular, for all u ∈ Ψ−1 2 (I0 ), we have ϕ1 (r) ≤

Φ(u) r − Ψ2 (u)

for all r ∈ (Ψ2 (u), 0).

Hence, (3.3.19) holds, where now ϕ1 (0) is given by (3.4.14). Also in this setting (3.3.20) and (3.3.21) hold and (3.3.22) simply reduces to ? |Ψ2 (u)| ≤ Cp? kukp . Taken r < 0 and v ∈ Ψ−1 2 (r), we obtain r = Ψ2 (v) ≥ −C kvk p?

p?

≥ −C

p p?

c

p? /p Φ(v) ,

where in the last step we have used (A)–(c). Therefore, by (3.2.2), since r u ≡ 0 ∈ Ψ−1 2 (I ), ϕ2 (r) ≥ −p/p?

where κ = c Cp?

1 |r|

inf −1

? −1

Φ(v) ≥ κ|r|p/p

,

v∈Ψ2 (r)

/p. This implies that lim− ϕ2 (r) = ∞, being p? > p. r→0

The conclusion follows exactly as in the proof of Corollary 3.3.8.

2

It is apparent from the main definitions (3.4.9), (3.4.10), Corollary 3.4.9 and (3.4.14) that 0 < λ? < `? ≤ `? ≤ λ? . Hence Corollary 3.4.9 provides also the useful information that 0 < λ? < λ? . Existence of Three Solutions for Robin Eigenvalue Problems Involving a Divergence Type Operator In this subsection we study problem (3.1.5) in a C 1 bounded domain Ω of Rn , 1 < p < n, n ≥ 2 and assume that w is a weight in Ω of class

64


L$ (Ω), with $ > n/p. The trace weight w1 is assumed to belong to the space L(n−1)/(p−1) (∂Ω), where p∗ =

(n − 1)p n−p

is the critical exponent for the traces, and there exists a positive constant w0 such that w1 (x) ≥ w0 > 0 for all x ∈ ∂Ω. In this setting we consider for X the Sobolev space W 1,p (Ω), endowed with the usual norm 1/p kukW 1,p (Ω) = kDukpp + kukpp . Furthermore, by Theorem 4.26 of [1] there exists a linear extension operator E : W 1,p (Ω) → W 1,p (Rn ), i.e. for all u ∈ W 1,p (Ω) (i) Eu Ω = u; (ii) kEukLp (Rn ) ≤ Kkukp ; (iii) kEukW 1,p (Rn ) ≤ KkukW 1,p (Ω) , where k·kW 1,p (Rn ) denotes the standard norm in W 1,p (Rn ). The constant K depends only on Ω. Hence, the embedding W 1,p (Ω) ,→ Lp∗ (∂Ω) is continuous by Theorem 5.22 of [1]. Moreover, by Hölder’s inequality, Lp∗ (∂Ω) ,→ Lp (∂Ω; w1 ) continuously, indeed Z w1 (x)|u|p dS ≤ kw1 k(n−1)/(p−1),∂Ω kukpp∗ ,∂Ω for all u ∈ Lp∗ (∂Ω). ∂Ω

This implies that also the embedding W 1,p (Ω) ,→ Lp (∂Ω; w1 ) is continuous. Lemma 3.4.10. The function k · k : W 1,p (Ω) → R+ 0, 1/p Z Z p p kuk = |Du| dx + w1 (x)|u| dS , Ω

∂Ω

is well defined and is an equivalent norm on W 1,p (Ω).

65

Proof. Clearly, k · k is well defined in W 1,p (Ω), since the embedding W 1,p (Ω) ,→ Lp (∂Ω; w1 ) is continuous, as shown above. Following somehow the proof of Lemma 1 of [91], we first show that there exist two positive constants C1 and C2 such that for every u ∈ W 1,p (Ω) Z Z Z p p |u| dx ≤ C1 |Du| dx + C2 |u|p dS. (3.4.15) Ω

Ω

∂Ω

C0∞ (Rn )

we have Indeed, for all u ∈ Z Z Z p p x · D(|u| )dx = (ν · x)|u| dS − n |u|p dx. Ω

∂Ω

Ω

Hence, by the Hölder and Young inequalities, with ε = n/2(p − 1)CΩ and CΩ = maxx∈Ω |x|, we get Z Z Z p p |x| · |u| dS + p |x| · |u|p−1 |Du|dx n |u| dx ≤ Ω ∂Ω Ω (Z Z 0 Z ) 1/p

|u|p dS + p

≤ CΩ ∂Ω

Z ≤ CΩ

Ω

Z

|u| dS + (p − 1)ε ∂Ω

|Du|p dx

Ω p

1/p

|u|p dx

p

|u| dx + ε Ω

1−p

Z

p

|Du| dx . Ω

Therefore, (3.4.15) is valid for all u ∈ C0∞ (Rn ), with the constants C1 = (2CΩ /n)p (p − 1)p−1 , C2 = 2CΩ /n. The desired inequality (3.4.15) holds in the entire W 1,p (Ω) by standard density arguments. Now, in force of (3.4.15), Z Z C2 p p kukW 1,p (Ω) ≤ (1 + C1 ) |Du| dx + w1 (x)|u|p dS ≤ C3 kukp , w 0 ∂Ω Ω where C3 = max{1 + C1 , C2 /w0 }. Conversely, the embedding W 1,p (Ω) ,→ Lp (∂Ω; w1 ) is continuous, as proved above, and denote by S?,w1 the Sobolev constant of this embedding. Hence, for all u ∈ W 1,p (Ω) Z Z p p w1 (x)|u|p dS kuk = |Du| dx + ∂Ω ZΩ p ≤ |Du|p dx + S?,w kukpW 1,p (Ω) ≤ C4 kukpW 1,p (Ω) , 1 Ω

66


2

p where C4 = 1 + S?,w . This concludes the proof. 1

In the proof of Lemma 3.4.10 one can take for ε every number in the open interval (0, n/(p − 1)CΩ ). From now on we endow the space W 1,p (Ω) with the norm k · k defined in Lemma 3.4.10. Lemma 3.4.11. Let A satisfy (A) from (a) to (c). Then the functional Φ : W 1,p (Ω) → R, defined by Φ(u) = Φ1 (u) + Φ2 (u), Z 1 A (x, Du(x))dx, Φ2 (u) = w1 (x)|u|p dS, Φ1 (u) = p Ω ∂Ω Z

is convex, weakly lower semicontinuous and of class C 1 (W 1,p (Ω)). Moreover, Φ0 : W 1,p (Ω) → [W 1,p (Ω)]? verifies the (S+ ) condition, i.e., for every sequence (uk )k ⊂ W 1,p (Ω) such that uk * u weakly in W 1,p (Ω) and lim suphΦ0 (uk ), uk − ui ≤ 0,

(3.4.16)

k→∞

then uk → u strongly in W 1,p (Ω). Proof. A simple calculation shows that the functional Φ is convex and of class C 1 (W 1,p (Ω)). Hence, in particular Φ is weakly lower semicontinuous on W 1,p (Ω), see Corollary 3.9 of [23]. Similarly, Φ1 and Φ2 are convex and of class C 1 (W 1,p (Ω)) and so weakly lower semicontinuous on W 1,p (Ω). Let (uk )k be a sequence in W 1,p (Ω) as in the statement. Then Φi (u) ≤ lim inf Φi (uk ), being Φi weakly lower semicontinuous on W 1,p (Ω), i = 1, 2. R Furthermore, Ω A(x, Du) · (Duk − Du)dx → 0 as k → ∞, since in 0 particular RDuk * Du in [Lp (Ω)]n , and |A(x, Du)| ∈ Lp (Ω) by (A)–(c). Similarly, ∂Ω w1 (x)|u|p−2 u(uk − u)dS → 0, being uk * u in Lp (∂Ω; w1 ) 0 and clearly |u|p−2 u ∈ Lp (∂Ω; w1 ). Hence, by convexity and (3.4.16), we

67

have at once Z 0 ≤ lim sup

(A(x, Duk ) − A(x, Du)) · (Duk − Du)dx Z p−2 p−2 + w1 (x)(|uk | uk − |u| u)(uk − u)dS ∂Ω Z 0 = lim sup hΦ (uk ), uk − ui − A(x, Du) · (Duk − Du)dx k→∞ Ω Z p−2 w1 (x)|u| u(uk − u)dS ≤ 0. − k→∞

Ω

∂Ω

Therefore, again by convexity, for all i = 1, 2 lim hΦ0i (uk ) − Φ0i (u), uk − ui = lim hΦ0i (uk ), uk − ui = 0,

k→∞

k→∞

lim Φi (uk ) = Φi (u).

(3.4.17)

n→∞

In particular, up to a subsequence, Duk → Du a.e. in Ω and uk → u a.e. on ∂Ω by Lemma 3.2.3, with N = n − 1, A = ∂Ω, s = p, ω = w1 and dx = dS. Since (3.4.6) holds for Φ1 thanks to (3.4.17), following the proof of Lemma 3.3.2 we still get that limk→∞ kDuk − Dukp = 0. Similarly, since (3.4.6) holds for Φ2 thanks to (3.4.17), we consider the sequence (hk )k in L1 (∂Ω; w1 ) defined pointwise by p |uk (x)|p + |u(x)|p uk (x) − u(x) hk (x) = − . 2 2 Thus, by convexity hk ≥ 0 and hk (x) → |u(x)|p for a.a. x ∈ ∂Ω. Therefore, multiplying by w1 (x), we have in force of the Fatou lemma and (3.4.17) Z 1 p Φ2 (u) ≤ lim inf w1 (x)hk (x)dS = p Φ2 (u)− p lim sup kuk −ukpp,w1 ,∂Ω . k→∞ 2 k→∞ ∂Ω Hence, lim supk→∞ kuk −ukp,w1 ,∂Ω ≤ 0, that is limk→∞ kuk −ukp,w1 ,∂Ω = 0. In conclusion, limk→∞ kuk − uk = 0, as required. 2 We introduce the first eigenvalue λ1 of the problem ( −∆p u = λw(x)|u|p−2 u in Ω, p−2 ∂u p−2 |Du| ∂ν + w1 (x)|u| u = 0 on ∂Ω,

(3.4.18)

68


that is R λ1 =

inf

u∈W 1,p (Ω) u6=0

R p |Du| dx + w (x)|u|p dS Ω ∂Ω 1 R . w(x)|u|p dx Ω

(3.4.19)

Furthermore, we have the main property. Theorem 3.4.12. The infimum λ1 in (3.4.19) is attained, that is there exists u1 ∈ W 1,p (Ω), with ku1 kp,w = 1, which realizes the minimum in (3.4.19) and represents an eigenfunction for (3.4.18) corresponding to λ1 . Moreover, λ1 > 0 and λ1 kukpp,w ≤ kukp

for every u ∈ W 1,p (Ω).

(3.4.20)

Proof. First, I(u) = kukp and J (u) = kukpp,w are continuously Gâteaux differentiable, convex in W 1,p (Ω), and clearly I 0 (0) = J 0 (0) = 0. Moreover, J 0 (u) = 0 implies u = 0. In particular, I and J are weakly lower semicontinuous on W 1,p (Ω). Actually, J is weakly sequentially continuous on W 1,p (Ω). Indeed, if (uk )k ⊂ W 1,p (Ω) and uk * u in W 1,p (Ω), then uk → u in Lp (Ω; w), being the natural embedding between W 1,p (Ω) and Lp (Ω; w) compact, since w ∈ L$ (Ω), with $ > n/p. This implies at once that J (uk ) = kuk kpp,w → J (u) = kukpp,w , as claimed. Clearly, I is coercive in W 1,p (Ω), so that, if W = {u ∈ W 1,p (Ω) : J (u) = kukpp,w ≤ 1} is unbounded with respect to the norm k · k, clearly I is coercive also in W , otherwise no further assumptions are required to I. In conclusion, all the hypotheses of Theorem 6.3.2 of [18] are fulfilled and so λ1 is attained in {u ∈ W 1,p (Ω) : J (u) = kukpp,w = 1}. In other words, there exists u1 ∈ W 1,p (Ω), with ku1 kp,w = 1, which realizes the minimum in (3.4.19) and represents an eigenfunction for (3.4.18) corresponding to λ1 . Therefore, (3.4.20) holds. 2 Throughout this subsection we assume (A) from (a) to (c) and that 0 < c ≤ 1 ≤ C in (c). On the perturbation f we require (F)–(a), (b) and (c1 ), where now u1 is the first normalized eigenfunction of (3.4.18). As in Section 3.3, conditions (F)–(a) and (b) imply that f (x, 0) = 0 for a.a. x ∈ Ω, and that (3.3.5)–(3.3.8) hold. On g we suppose

69

(G) Let g : ∂Ω × R → R be a Carathéodory function, verifying the next properties. (a) There exist an exponent q˜ ∈ (1, p) and a function g0 ∈ Lβ (∂Ω), with β > (n − 1)/(p − 1), 0 ≤ g0 (x) ≤ Cg w1 (x) for a.a. x ∈ ∂Ω and some Cg > 0, such that |g(x, s)| ≤ g0 (x)|s|q˜−1 for all (x, s) ∈ ∂Ω × R. (b) Sg = ess sup s6=0 x∈∂Ω

|g(x, s)| ∈ R+ . p−1 w1 (x)|s|

Observe that (G)–(a) implies that g(x, 0) = 0 for a.a. x ∈ ∂Ω. Hence, the positive number λµ =

c λ1 p 1 + Sf + λ1 |µ|Sg S?,w 1

is well defined for all µ ∈ R, by (F)–(b) and (G)–(b), and S?,w1 is the best Sobolev constant of the continuous embedding W 1,p (Ω) ,→ Lp (∂Ω; w1 ); see the beginning of this subsection. The main result for this problem is proved by using the canonical energy functional Jλ,µ associated to (3.1.5), which is given by Jλ,µ (u) = Φ(u) + λ[Ψ(u) + µΥ(u)], where Ψ(u) = Ψ1 (u) + Ψ2 (u), Z 1 p Ψ1 (u) = − kukp,w , Ψ2 (u) = − F (x, u(x))dx, p Ω Z 1 Φ(u) = A (x, Du(x))dx + kukpp,w1 ,∂Ω , p Ω Z Z s Υ(u) = − G(x, u(x))dS, with G(x, s) = g(x, t)dt. ∂Ω

(3.4.21)

0

It is easy to see that the functional Jλ,µ is well defined in W 1,p (Ω) and

70


of class C 1 (W 1,p (Ω)). Furthermore, Z Z 0 w1 (x)|u|p−2 uϕdS hJλ,µ (u), ϕi = A(x, Du(x))Dϕ(x)dx + Ω ∂Ω Z −λ w(x)|u(x)|p−2 u(x) + f (x, u(x)) ϕ(x)dx ZΩ +µ g(x, u)ϕdS , ∂Ω

where, h·, ·i denotes the duality pairing between W 1,p (Ω) and its dual space [W 1,p (Ω)]? . Therefore, the critical points u ∈ W 1,p (Ω) of the functional Jλ,µ are the weak solutions of problem (3.1.5). Using the notation of Theorem 3.2.2, if Ψ(v) < 0 at some v ∈ W 1,p (Ω), then the crucial positive number λ? = ϕ1 (0) =

inf −1

u∈Ψ

− (I0 )

Φ(u) , Ψ(u)

I0 = (−∞, 0),

is well defined, as in Section 3.3, and independent of µ. Lemma 3.4.13. Ψ−1 (I0 ) is non-empty. For all µ ∈ R we have c λµ ≤ λ1 = λ? ≤ λ? < cλ1 1 + Sf and λµ < λ? ≤ λ? if µ 6= 0. Proof. From (F)–(c1 ) it follows that C 1 1 C − 1 = − < 0, Ψ(u1 ) < − − p p c cp

i.e. u1 ∈ Ψ−1 (I0 ),

being C ≥ c > 0 by (A)–(c). Hence λ? is well defined. Again by (3.4.1) and (F)–(c1 ) Φ(u) Φ(u1 ) λ? = ϕ1 (0) = inf − ≤ − u∈Ψ−1 (I0 ) Ψ(u) Ψ(u1 ) R R p Ω A (x, Du1 )dx + ∂Ω w1 (x)|u1 |p dS = −pΨ(u1 ) R R p C Ω |Du1 | dx + ∂Ω w1 (x)|u1 |p dS < ≤ c ku1 kp = c λ1 , C/c

71

since here 1 ≤ C by assumption. Finally, by (3.4.1), (3.4.21), (3.3.8) and (3.4.19), for all u ∈ W 1,p (Ω), with u 6= 0, we have ckDukpp + kukpp,w1 ,∂Ω Φ(u) ckukp c ≥ ≥ ≥ λ1 = λ? ≥ λµ . p p |Ψ(u)| (1 + Sf )kukp,w (1 + Sf )kukp,w 1 + Sf since c ≤ 1 and µ could be zero. Hence, in particular λ? ≥ λ? ≥ λµ .

2

Lemma 3.4.14. The operators Ψ01 , Ψ02 , Ψ0 : W 1,p (Ω) → [W 1,p (Ω)]? are compact and Ψ1 , Ψ2 , Ψ are sequentially weakly continuous in W 1,p (Ω). Proof. We show that Ψ01 and Ψ02 are compact using exactly the same arguments produced for the proof of Lemma 3.4.5, with the only difference that here the space W 1,p (Ω) takes the place of W01,p (Ω). Hence, Ψ0 = Ψ01 + Ψ02 is compact and Ψ is sequentially weakly continuous by [111, Corollary 41.9], being W 1,p (Ω) reflexive. 2 Remark 3.4.15. The embedding W 1,p (Ω) ,→ Lh (∂Ω) is compact for all h ∈ (p, p∗ ). Indeed, by Theorem 1.4.3.2 of [58] it results that the embedding W 1,p (Ω) ,→ W 1−ε,p (Ω) is compact for every ε ∈ (0, 1). Now, Theorem 7.58–(i) of [1], with s = 1−ε, k = n−1, q = h, χ = s−(n/p)+(k/q) = 0, ensures that the embedding W 1−ε,p (Ω) ,→ Lh (∂Ω) is continuous. Hence, if we take ε = 1 − (n/p) + [(n − 1)/h], we have χ = 0 and ε ∈ (0, 1), being h ∈ (p, p∗ ). This proves the claim. Lemma 3.4.16. The operator Υ0 : W 1,p (Ω) → [W 1,p (Ω)]? is compact. Proof. Since Υ0 is continuous and W 1,p (Ω) is a reflexive Banach space, it is enough to show that Υ0 is weak-to-strong sequentially continuous, i.e. if uk * u in W 1,p (Ω), then kΥ0 (uk ) − Υ0 (u)k? → 0 as k → ∞. To this aim, fix (uk )k ⊂ W 1,p (Ω), with uk * u in W 1,p (Ω). By the Remark 3.4.15 and the facts that β > p∗ and 1 < q˜ < p, the embedding 0 0 W 1,p (Ω) ,→ Lβ q˜(∂Ω) is compact. Moreover, Lβ q˜(∂Ω) ,→ Lq˜(∂Ω; g0 ) continuously, being g0 ∈ Lβ (∂Ω) by (G)–(a). Therefore, the embedding W 1,p (Ω) ,→ Lq˜(∂Ω; g0 ) is compact and so uk → u in Lq˜(∂Ω; g0 ).

72

Chapter 3. Eigenvalue Systems 1/(1−˜ q)

0

Define Ng : Lq˜(∂Ω; g0 ) → Lq˜ (∂Ω; g0 ) by Ng (v) = g(·, v(·)) for 0 1/(1−˜ q) q˜ ) all v ∈ L (∂Ω; g0 ). We assert that Ng (uk ) → Ng (u) in Lq˜ (∂Ω; g0 as k → ∞. Indeed, fix a subsequence (ukj )j of (uk )k . Hence, there exists a subsequence, still denoted by (ukj )j , satisfying (i) and (ii) of Lemma 3.2.3, with N = n−1, A = ∂Ω, ω = g0 and s = q˜. Thus, ukj → u a.e. in ∂Ω (in the sense of the n − 1 Lebesgue measure on ∂Ω), and |ukj | ≤ h a.e. in ∂Ω for all k ∈ N and some h ∈ Lq˜(∂Ω; g0 ). In particular, 0 1/(1−˜ q) |Ng (ukj )−Ng (u)|q˜ g0 → 0 a.e. in ∂Ω, being g(x, ·) continuous for a.a. 0 1/(1−˜ 0 q) x ∈ ∂Ω. Furthermore, |Ng (ukj ) − Ng (u)|q˜ g0 ≤ 2q˜ g0 hq˜ ∈ L1 (∂Ω). Hence, by the dominated convergence theorem, we have Ng (ukj ) → Ng (u) 0 1/(1−˜ q) ). Therefore the entire sequence (Ng (uk ))k converges to in Lq˜ (∂Ω, g0 1/(1−˜ q) q˜0 Ng (u) in L (∂Ω, g0 ) as k → ∞, as asserted. Finally, for all ϕ ∈ W 1,p (Ω), with kϕk = 1, we have by Hölder’s inequality,

0

0

Z

|hΥ (uk ) − Υ (u), ϕi| ≤

−1/˜ q

g0

1/˜ q

|Ng (uk ) − Ng (u)| · g0 |ϕ|dS

∂Ω

≤ kNg (uk ) − Ng (u)kq˜0,g1/(1−˜q) ,∂Ω kϕkq˜,g0 ,∂Ω 0

≤ S?,˜q,g0 kNg (un ) − Ng (u)kq˜0,g1/(1−˜q) ,∂Ω , 0

where S?,˜q,g0 is the Sobolev constant for the embedding W 1,p (Ω) ,→ Lq˜(∂Ω; g0 ). Thus, kΥ0 (uk ) − Υ0 (u)k? → 0 as k → ∞, that is Υ0 is compact. 2

Lemma 3.4.17. The functional Jλ,µ (u) = Φ(u) + λ[Ψ(u) + µΥ(u)] is coercive for every λ ∈ I = (−∞, cλ1 ) and µ ∈ R.

Proof. Fix λ ∈ I and µ ∈ R. Then by (A)–(c), (F)–(a), (G)–(a) and

73

(3.4.19) Z Z c λ p p Jλ,µ (u) ≥ kuk − kukp,w − |λ| |F (x, u)|dx − |λ| · |µ| |G(x, u)|dS p p Ω ∂Ω Z 1 λ p ≥ c− kuk − |λ| f0 (x)dx p λ1 Ω Z Z f1 (x) |λ| · |µ| q f0 (x) + − |λ| |u| dx − g0 (x)|u|q˜dS q q ˜ ∂Ω Ω λ 1 c− kukp − |λ|C1 − |λ| C2 kukq − |λ| · |µ|C3 kukq˜ ≥ p λ1 q where C1 = kf0 k1 , C2 = S$ q and S$0 q 0 q kf0 +f1 /qk$ , C3 = S?,β 0 q ˜kg0 kβ,∂Ω /˜ 0 1,p denotes the best Sobolev constant of the embedding W (Ω) ,→ L$ q (Ω), 0 while S?,β 0 q˜ is the Sobolev constant of the embedding W 1,p (Ω) ,→ Lβ q˜(∂Ω). Note that C1 < ∞, since f0 ∈ L$ (Ω) ⊂ L1 (Ω) by (F)–(a), being $ > 1 and Ω bounded. This concludes the claim, since q, q˜ ∈ (1, p). 2

Theorem 3.4.18. Assume (A) from (a) to (c). (i) If (F)–(a), (b) and (G)–(a), (b) hold, then for every µ ∈ R there exists λµ = c λ1 /(1 + Sf + λ1 |µ|Sg cp?,w1 ) such that problem (3.1.5) has only the trivial solution for all λ ∈ [0, λµ ). (ii) If conditions (F)–(a), (b), (c1 ) and (G)–(a), (b) hold, then for every compact interval [a, b], with [a, b] ⊂ (λ? , cλ1 ) and λ? = ϕ1 (0) < cλ1 , there exists δ > 0 such that problem (3.1.5) admits at least two nontrivial solutions for all λ ∈ [a, b] and µ ∈ (−δ, δ). Proof. (i) Let u ∈ W 1,p (Ω) be a weak solution of problem (3.1.5), then Z

Z

A(x, Du)Dϕdx + w1 (x)|u|p−2 uϕ dS Ω ∂Ω Z Z p−2 {w(x)|u| u + f (x, u)}ϕdx + µ =λ Ω

g(x, u)ϕ dS

∂Ω

for all ϕ ∈ W 1,p (Ω). Hence, by taking ϕ = u and using (A)–(c), (G)–(b)

74


and (3.3.6), we get Z Z p p cλ1 kuk ≤ λ1 A(x, Du)Du dx + w1 (x)|u| dS Ω ∂Ω Z Z p = λ1 λ {w(x)|u| + f (x, u)u} dx + µ g(x, u)u dS Ω ∂Ω Z |f (x, u)| p ≤ λ1 λ kukp,w + w(x)|u|p dx p−1 Ω w(x)|u| Z |g(x, u)| p w (x)|u| dS +|µ| p−1 1 ∂Ω w1 (x)|u| ≤ λ (1 + Sf ) kukp + λ1 |µ|Sg kukpp,w1 ,∂Ω ≤ λ 1 + Sf + λ1 |µ|Sg cp?,w1 kukp . This concludes the proof of part (i). (ii) By Lemma 3.4.11 the functional Φ is weakly lower semicontinuous and of class C 1 (W 1,p (Ω)) and Φ0 verifies the (S+ ) condition. Moreover, Ψ is weakly lower semicontinuous, continuously Gâteaux differentiable in W 1,p (Ω) and Ψ0 is compact by Lemma 3.4.14. Finally, Υ is continuously Gâteaux differentiable in W 1,p (Ω) and Υ0 is compact by Lemma 3.4.16. We claim that Ψ(W 1,p (Ω)) ⊃ R− 0 . Indeed, as shown in the proof of Theorem 3.3.7, Ψ(0) = 0 and 1 (p−q)/p Ψ(u) ≤ − kukpp,w + kf0 k1 + 2Cf kwk1 kukqp,w . p Therefore lim

u∈W 1,p (Ω) kukp,w →∞

Ψ(u) = −∞,

being 1 < q < p. Hence, the claim follows by the continuity of Ψ. Thus, (inf Ψ, sup Ψ) ⊃ R− 0 . Arguing as in the proof of Theorem 3.3.7, we see that (3.3.20)–(3.3.22) continue to hold for every u ∈ W 1,p (Ω), since the embedding W 1,p (Ω) ,→ Lγ (Ω; w) is continuous being p? ∈ (p, p∗ /$0 ). We recall that here k · k denotes the norm on W 1,p (Ω) defined in Lemma 3.4.10. Therefore, given r < 0 and v ∈ Ψ−1 (r), condition (A)–(c) yields (3.4.11), as before. Since the functional Φ is bounded below, coercive and lower semicontinuous on the reflexive Banach space

75

W 1,p (Ω), it is also coercive on the sequentially weakly closed non-empty set Ψ−1 (r), see Lemma 3.4.14. Therefore, again by Theorem 6.1.1 of [18], there exists an element ur ∈ Ψ−1 (r) such that Φ(ur ) = inf Φ(v). By −1 v∈Ψ

(r)

(3.2.2), we have ϕ2 (r) ≥ −

Φ(ur ) 1 inf Φ(v) = , −1 r v∈Ψ (r) |r|

being u ≡ 0 ∈ Ψ−1 (I r ). Hence, by (3.4.11) we get again the crucial inequality (3.4.12), so that either lim inf ϕ2 (r) ≥ cλ1 if ϕ2 is locally − r→0

bounded at 0− , or lim sup ϕ2 (r) = ∞. r→0−

Moreover, arguing as in the proof of Theorem 3.3.7, we see that (3.3.19) continues to hold, with Φ now defined as in Lemma 3.4.11. In particular, by (3.3.19) and Lemma 3.4.13, we obtain that for all integers k ≥ k ? = 1 + [2/(cλ1 − λ? )] there exists a number rk < 0 so close to zero that ϕ1 (rk ) < λ? + 1/k < cλ1 − 1/k < ϕ2 (rk ). Hence, by Theorem 3.2.2 for every compact interval ?

[a, b] ⊂ (λ , cλ1 ) =

∞ [ k=k?

?

[λ + 1/k, cλ1 − 1/k] ⊂

∞ [

(ϕ1 (rk ), ϕ2 (rk )) ∩ I

k=k?

there exists δ > 0 such that, being u ≡ 0 a critical point of Jλ,µ , problem (3.1.5) admits at least two nontrivial solutions for every λ ∈ [a, b] and µ ∈ (−δ, δ), as claimed. 2 According to the Remark 3.4.8, also in this context we can treat the new slightly easier problem ( −divA(x, Du) = λf (x, u) in Ω, (3.4.22) p−2 A(x, Du) · ν + w1 (x)|u| u = λµg(x, u) on ∂Ω, where f satisfies the conditions required in Corollary 3.3.8 and g assumptions (G)–(a) and (b). We have the following multiplicity result. Corollary 3.4.19. Assume (A) from (a) to (c), suppose that f : Ω×R → R satisfies (F)–(a), (b), and that g : ∂Ω × R → R verifies (G)–(a) and (b).

76


(i) For every µ ∈ R there exists `µ = c λ1 /(Sf + λ1 |µ|Sg cp?,w1 ) such that problem (3.4.22) has only the trivial solution for all λ ∈ [0, `µ ). (ii) If furthermore f satisfies (F)–(c0 ) and [a, b] ⊂ (`? , ∞) is a nondegenerate closed interval, with `? = ϕ1 (0), then there exists δ > 0 such that problem (3.4.22) has at least two nontrivial solutions for every λ ∈ [a, b] and µ ∈ (−δ, δ). Proof. The energy functional Jλ,µ associated to (3.4.22) is given by Jλ,µ (u) = Φ(u) + λ[Ψ2 (u) + µΥ(u)], where Z Ψ2 (u) = − F (x, u(x))dx. Ω

First, note that Jλ,µ is coercive for every λ, µ ∈ R. Indeed, as shown in the proof of Lemma 3.4.17 Z Z c c p |G(x, u)|dS ≥ kukp Jλ,µ (u) ≥ kuk − |λ| |F (x, u)|dx − |λ| · |µ| p p Ω ∂Ω q q˜ − |λ|C1 − |λ| C2 kuk − |λ| · |µ|C3 kuk q where C1 = kf0 k1 , C2 = S$ q and c$0 q 0 q kf0 +f1 /qk$ , C3 = S?,β 0 q ˜kg0 kβ,∂Ω /˜ 0 1,p denotes the best Sobolev constant of the embedding W (Ω) ,→ L$ q (Ω), 0 while S?,β 0 q˜ is the Sobolev constant of the embedding W 1,p (Ω) ,→ Lβ q˜(∂Ω). This shows the claim, since q, q˜ ∈ (1, p). Hence, here I = R. The part (i) of the statement is proved using the same argument produced for the proof of Theorem 3.4.18–(i), being Z Z p p c λ1 kuk ≤ λ1 A(x, Du)Du dx + w1 (x)|u| dS Ω ∂Ω Z Z g(x, u)u dS f (x, u)u dx + µ = λ1 λ ∂Ω Ω ≤ λ1 λ Sf kukpp,w + |µ|Sg kukpp,w1 ,∂Ω ≤ λ Sf + λ1 |µ|Sg cp?,w1 kukp .

Thus, if u is a nontrivial weak solution of (3.4.22), then necessarily λ ≥ `µ , as required. If there exists u0 ∈ W 1,p (Ω) such that Ψ2 (u0 ) < 0, then the crucial number Φ(u) `? = ϕ1 (0) = inf − −1 Ψ2 (u) u∈Ψ2 (I0 )

77

is well defined. The function u0 ∈ C0 (Ω), produced in the proof of Corollary 3.3.8, has the property that ku0 kp,w1 ,∂Ω = 0, so that u0 ∈ (W 1,p (Ω), k · k) and Ψ2 (u0 ) < 0 for σ ∈ (0, 1) sufficiently large, as before. From now on the argument can proceed exactly as the proof of Corollary 3.3.8 and concluded by using Theorem 3.2.2. Indeed, (3.3.27) now shows that for all integers k ≥ k ? = 2 + [`? ] there exists rk < 0 so close to zero that ϕ1 (rk ) < `? + 1/k < k < ϕ2 (rk ). Hence, since I = R, by Theorem 3.2.2 for every compact interval ?

[a, b] ⊂ (` , ∞) =

∞ [ k=k?

?

[` + 1/k, k] ⊂

∞ [

(ϕ1 (rk ), ϕ2 (rk ))

k=k?

there exists δ > 0 such that, being u ≡ 0 a critical point of Jλ,µ , problem (3.4.22) admits at least two nontrivial solutions for every λ ∈ [a, b] and µ ∈ (−δ, δ), as claimed. 2

Part II

Damped Evolution Kirchhoff Systems

Chapter 4

Global Non-continuation and Qualitative Analysis for p(x)-Polyharmonic Kirchhoff Systems In this chapter we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time dependent nonlinear dissipative and driving forces.

4.1

Introduction

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff-Love theory for thin plates subjected to forces and moments. More precisely, we are interested in global non-existence and a priori estimates for the lifespan of maximal solutions, where for lifespan we mean the supremum of all t’s for which the solution exists. For L = 1, 2, . . . the system we analyze in R+ 0 × Ω is  u + M (IL (u)) ∆Lp(x) u + N (IL−1 (u)) ∆L−1  p(x) u  tt  + µ|u|p(x)−2 u + Q(t, x, u, ut ) = f (t, x, u), (4.1.1)   α  D u(t, x) =0 for all α, with |α| ≤ L − 1, + R0 ×∂Ω

82

Chapter 4. Global Non-continuation

where Ω ⊂ Rn is a bounded domain with Lipschitz boundary, the function u = (u1 , . . . , ud ) = u(t, x) is the vectorial displacement, d ≥ 1, R+ 0 = [0, ∞), α is a multi-index, µ is a non-negative parameter and p ∈ C+log (Ω) is such that, in the notation of Section 1.1, either p+ < n/L or p− ≥ n/L. L,p(·) For L = 0, 1, 2, . . . the Dirichlet functional IL : [W0 (Ω)]d → R+ 0 is defined in (2.1.2). Since here u = u(t, x), depends also on t, we write IL (u(t)) =

Z Ω

|DL u(t, x)|p(x) dx, p(x)

where DL u = (DL u1 , . . . , DL ud ), DL ui for i = 1, . . . , d is given in (1.2.2), hence DL u is a d-vector if L is even, it is an nd-vector if L is odd. Therefore, also | · | stands for the d-Euclidean norm, if L is even, while it denotes the nd-Euclidean norm when L is odd. For L = 0, 1, 2, . . . the p(x)-polyharmonic operator and the vectorial p(x)-polyharmonic operator ∆Lp(x) have been introduced in (2.1.3). The p(x)-Laplace operators possess more complicated nonlinearities than p-Laplace operators, mainly due to the fact that they are not homogeneous. Problem (4.1.1), in a simplified form, has its origin in the canonical model introduced by Woinowsky-Krieger, which arises in the dynamic buckling of a hinged extensible beam, subjected to an axial force. The main Kirchhoff function M is non trivial, that is M 6≡ 0, and satisfies (M ) M ∈ L1loc (R+ 0 ) is non-negative and there exists γ ≥ 1 such that Z τ + γM (τ ) ≥ τ M (τ ), τ ∈ R0 , M (z)dz. where M (τ ) = 0

The second Kirchhoff function N , possibly trivial, verifies condition (M ), with parameter η ∈ [1, γ], that is ηN (τ ) ≥ τ N (τ ) for all τ ∈ R+ 0 , where Rτ N (τ ) = 0 N (z)dz. d d d The nonlinear damping Q ∈ C(R+ 0 × Ω × R × R → R ) satisfies (Q(t, x, u, v), v) ≥ 0

for all t, x, u, v,

(4.1.2)

83

Figure 4.1: The phase plane d d and f ∈ C(R+ 0 × Ω × R → R ) is derivable from a potential F , that is

f (t, x, u) = Du F (t, x, u),

F (t, x, 0) = 0.

(4.1.3)

From now on, without further mentioning, we assume (4.1.2), (4.1.3) and the structure conditions on M and N described above. The term Q represents the most common suppressions of the vibrations of an elastic structure of passive viscous type and absorbs vibration energy, while f is an external source force. It is worth noting that the dissipative function Q makes easier the existence of global solutions for (4.1.1), since Q plays the role of a stabilizing term. In contrast, the source force f alone can drive the solution to blow up in finite time, since it works against the continuation of local solutions. The key result is the general Theorem 4.3.1 for (4.1.1), in which we prove that no global solutions u of the problem can exist, whenever the initial energy Eu(0) is controlled above by a critical value E2 , see Figure 4.1. The proof relies on concavity arguments combined with a new version of the potential well method, following a pioneering idea of Pucci and Serrin contained in [97], see also [15] for p(x)-Kirchhoff systems, where N = 0 and Q is less general. The values E1 and E2 in Figure 4.1 depend on the solution u, so that concrete applications of Theorem 4.3.1 are given in Theorems 4.5.7, 4.6.1 and in their corollaries, in special subcases of f , M and Q, under the main restriction Eu(0) ≤ E0 , since the value E0 depends only on the geometry of the system (4.1.1) and not on

84


the specific solution u. In particular, Q can be taken of the form Q(t, x, u, v) = d1 (t, x)|u|κ |v|m−2 v + d2 (t, x, u)|v|℘−2 v, where d1 , d2 are non-negative functions and κ, m and ℘ are non-negative exponents satisfying appropriate relations. More precisely, m describes the growth of Q for v small in magnitude and ℘ is the growth of Q for v large in magnitude, see Section 4.4 for further details. The applications of Section 4.5 are obtained thanks to a deep qualitative analysis on (4.1.1) and its intrinsic geometric structure. As it is well known in literature for similar more standard problems, also in this generality we are able to prove that the energy of the system (4.1.1) lies always above a curve ϕ in the phase plane. The function ϕ attains its maximum value E0 at a point υ0 , see Figure 4.1 and (4.5.10)–(4.5.11). Moreover, a crucial role in this investigation and in the proofs of Theorems 4.5.7 and 4.6.1 is played by Lemmas 4.5.3 and 4.5.5, in which we establish fundamental properties of the solutions of (4.1.1), when the initial data u0 = u(0, ·) and u1 = ut (0, ·) verify the main restrictions ku0 kq(·) > υ0 and Eu(0) ≤ E0 , where q is a variable exponent related to the growth at infinity of f in u. The proofs of global non-existence in the literature do not, in general, imply finite time blow up of solutions. Recently, Guedda and Labani have proved in [59] a global non-existence theorem for a subcase of (4.1.1), when L = 2, p ≡ 2, M ≡ 1, µ = 0 and Q is linear, under dynamic boundary conditions. They assert that non-existence occurs by blow up, but the only sure thing deriving from the proof is that a non-continuation result is valid, not necessarily due to the blow up of solutions. A similar situation appears in [83] for a higher order Kirchhoff equation, where a finite upper bound T0 for the lifespan T of local solutions is obtained, without an explicit proof of blow up at the time T . As remarked in passing in [72], and further discussed by Ball in [16], it is possible for the solution to leave the domain of one of the differential operators in the equations before becoming unbounded. On the other hand, if one can couple the global non-existence with a local continuation argument, based on the assumption of an appropriate a priori bound for the solution, as in [56], then global non-existence will imply finite time blow up. The limit case Eu(0) = E0 , first studied by Vitillaro in [104], is covered here also when Q ≡ 0, provided that f satisfies the new assumption

85

(D)–(ii), namely when f depends effectively on t, see Section 4.5 for details. However, Vitillaro, under the more familiar hypothesis (D)–(i) (see Section 4.5 of this thesis), uses in [104] a different proof technique and considers only strong solutions, even if not explicitly stated, cf. [104, the proof of case (a) of Theorem 3, see also Section 4.4], as well as the comments in Section 2.2. The condition Eu(0) ≤ E0 has been assumed recently also by Chen and Zhou in [29], to prove the global non-existence of solutions of a semilinear Petrovsky equation, that is a subcase of (4.1.1), when L = 2, p ≡ 2, M ≡ 1, N ≡ 0 and µ = 0, and f and Q are of very special types. Moreover, for similar models, studied under dynamic boundary conditions and negative initial energy, we refer to [92], where non continuation results are given, recently extended to a more general setting and also improved in [12, 11]. Theorem 4.5.7 generalizes in several directions some results contained in Section 4.4 of [104], in Theorem 3.1 of [105] where L = 1, p ≡ 2, M ≡ 1, N ≡ 0 and the damping is linear in v, as well as in Theorem 2.1 of [29]. Finally, Theorem 4.5.7 is more general than Theorem 2.2 of [83] too. Theorem 4.6.1 establishes an a priori upper bound T0 for the lifespan T of maximal solutions of (4.1.1), only in terms of initial data. As far as we know, [9] is the first paper in which an a priori upper bound is explicitly determined for the nonlinear general problem (4.1.1). Simpler expressions for T0 are obtained when Q has a special form as in Corollary 4.6.4, which extends the recent Theorems 4.3 and 4.5 of [107]. More precisely, [107] deals with the subcase of (4.1.1) in which L = 1, p ≡ 2, N ≡ 0, µ = 0 and f and Q are power functions. About global non-existence and blow up of solutions for some nonlinear Kirchhoff equations included in the model (4.1.1), we should also mention Ono, and in particular the papers [88]–[90] and references therein. Finally, for related topics on existence and asymptotic behaviour of solutions for special subcases of (4.1.1) in which L = 2, p ≡ 2, M ≡ 1, Q = Q(v) and f = f (u), see [26, 27, 66] and references therein. The results of this chapter are an extension of the theorems of Sections 3–6 of [9] to the anisotropic case.

86


4.2

Preliminaries

In this section we collect a series of notations and preliminaries used throughout the chapter. Since we are in the vectorial setting, we consider maps assuming values in Rd , endowed with the Euclidean norm | · |d . As L,p(·) already said in Section 1.3, we endow the space [W0 (Ω)]d with the equivalent norm k · k = kDL · k[Lp(·) (Ω)]s ,

( d, s= nd,

if L = 2j, if L = 2j + 1.

For simplicity in notation we drop the exponents d and nd in all the funcL,p(·) L,p(·) tional spaces involved in the treatment, thus W0 (Ω) denotes [W0 (Ω)]d , and Lp(·) (Ω) = (Lp(·) (Ω), k · kp(·) ) is used in all the dimensions 1, d and nd. The main solution and test function space for (4.1.1) is L,p(·)

X = C(I → W0

(Ω)) ∩ C 1 (I → Lp(·) (Ω)),

where I = [0, T ) and T ∈ (0, ∞] is the lifespan of the maximal solution, namely it is the supremum of all t’s for which the solution exists, as already said in the introduction of this chapter. Fix now φ ∈ X. The functional A given by (A )

A φ(t) = M (IL (φ(t))) + N (IL−1 (φ(t))) + µ

Z Ω

|φ(t, x)|p(x) dx p(x)

includes the elliptic part of the system, and A φ is the Fréchet potential, with respect to φ, of the operator Aφ, defined pointwise for all (t, x) ∈ I × Ω by Aφ(t, x) = M (IL (φ(t)))∆Lp(x) φ(t, x) + N (IL−1 (φ(t)))∆L−1 p(x) φ(t, x) + µ|φ(t, x)|p(x)−2 φ(t, x). Clearly A φ ≥ 0 in I, being M and N non-negative and µ ≥ 0. Moreover

87

for all u, φ ∈ X and t ∈ I we have Z hAu(t, ·), φ(t, ·)i =M (IL (u(t))) |DL u(t, x)|p(x)−2 DL u(t, x)DL φ(t, x) dx Ω Z + N (IL−1 (u(t))) |DL−1 u(t, x)|p(x)−2 DL−1 u(t, x)· Ω Z · DL−1 φ(t, x) dx + µ |u(t, x)|p(x)−2 u(t, x)φ(t, x) dx, Ω

where the operation between DL u and DL φ (resp. between DL−1 u and DL−1 φ) is the d-Euclidean scalar product when L is even (resp. L − 1 is even), while it is the nd-Euclidean scalar product when L is odd (resp. L − 1 is odd). In particular, hAφ(t, ·), φ(t, ·)i = M (IL (φ(t)))ρp(·) (DL φ(t, ·)) + µρp(·) (φ(t, ·)) + N (IL−1 (φ(t)))ρp(·) (DL−1 φ(t, ·)), so that hAφ(t, ·), φ(t, ·)i ≤ γp+ A φ(t) for all (t, φ) ∈ I × X,

(4.2.1)

by the fact that M and N verify (M ) with parameters γ and η respectively and η ∈ [1, γ]. The basic assumption on f is (F1 )

F (t, ·, φ(t, ·)), (f (t, ·, φ(t, ·)), φ(t, ·)) ∈ L1 (Ω) for all t ∈ I, hf (t, ·, φ(t, ·)), φ(t, ·)i ∈ L1loc (I),

required along every φ ∈ X, where theR symbol (·, ·) denotes the Euclidean inner product of Rd and hϕ, ψi = Ω (ϕ(x), ψ(x))dx is the elementary bracket pairing, which is well defined for all ϕ, ψ such that (ϕ, ψ) ∈ L1 (Ω). The potential energy of the field φ ∈ X is given by Z F φ(t) = F (t, φ) = F (t, x, φ(t, x)) dx, Ω

and it is well defined by (F1 ). Denoting by Ft the partial derivative with L,p(·) respect to t of F = F (t, w) for (t, w) ∈ R+ (Ω), we assume the 0 × W0 following monotonicity condition (F2 )

L,p(·)

Ft ≥ 0 in R+ 0 × W0

(Ω).

88


The natural total energy of the field φ ∈ X, associated to (4.1.1), is the function Eφ(t) = 21 kφt (t, ·)k22 + A φ(t) − F φ(t),

(E )

and it is well defined in X by (F1 ). Inspired by [95] (see also [13]), we say that a (weak maximal) solution of (4.1.1) is a function u ∈ X satisfying the two properties (A) and (B) below: (A) Distribution Identity Z tn o t hut , φi 0 = hut , φt i − hAu, φi − hQ(τ, ·, u, ut ), φi + hf (τ, ·, u), φi dτ 0

for all t ∈ I and φ ∈ X; (B) Energy Conservation DuZ ∈ L1loc (I), t (ii) Eu(t) ≤ Eu(0) − Du(τ )dτ (i)

for all t ∈ I,

0

where the operator Du(t) = hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i +Ft u(t) describes the dynamic part of the system along the solution u ∈ X. Clearly Du ≥ 0 in I by (4.1.2) and (F2 ). The Distribution Identity is meaningful provided that hf (t, ·, u), φi ∈ 1 Lloc (I) and hQ(t, ·, u, ut ), φi ∈ L1loc (I), along the field φ ∈ X. The first condition is valid whenever (F1 ) is in charge, while the latter is assumed. These restrictions are satisfied in the special cases treated in the theorems, as well as in the applications. The other terms in the Distribution Identity are well defined thanks to the choice of the space X. In general it is important to consider weak solutions instead of strong solutions, namely functions u ∈ X satisfying (A), (B)–(i), with (B)–(ii) replaced by the Strong Energy Conservation, that is Z t (B)s (ii) Eu(t) = Eu(0) − Du(τ )dτ for all t ∈ I. 0

89

The main reason was first given in [95, Remark 4 at page 199]; see also [96, Remark 2 at page 49] and the discussion in [74, page 345]. Of course, if u is a strong solution, then Eu is non-increasing in I and this makes the analysis much simpler, see [29], [104]–[107], [109]. Concerning the external force f , beyond (F1 ) and (F2 ), we consider a third natural condition, which establishes a connection between the elliptic part A and the external source force f of the system. (F3 ) There exists a function q ∈ C+ (Ω) such that q− > max{γp+ , 2},

(4.2.2)

and for all F > 0 and φ ∈ X for which inf t∈I F φ(t) ≥ F, it results hf (t, ·, φ(t, ·)), φ(t, ·)i ≥ q− F φ(t) for all t ∈ I. In what follows, given a solution u ∈ X of (4.1.1), we put for convenience w1 = inf A u(t), t∈I γp+ E1 = 1 − w1 , q−

w2 = inf F u(t), t∈I q− − 1 w2 . E2 = γp+

(4.2.3)

Proposition 4.2.1. Let u ∈ X be a solution of (4.1.1), then A u and F u are bounded below in I. Proof. If u ∈ X is a solution of (4.1.1), then w1 ≥ 0 by (A ). Moreover, by (E ), (B)–(ii) and the fact that Du is non-negative, we get F u(t) ≥ w1 − Eu(0) ≥ −Eu(0) for all t ∈ I, so that w2 ≥ −Eu(0) > −∞.

2

Proposition 4.2.2. If u is a solution of (4.1.1) and Eu(0) < $w2 , with $ > −1, then w2 > 0. In particular, Eu(0) < E2 implies w2 > 0. Proof. Let u ∈ X be a solution of (4.1.1). As shown in the proof of Proposition 4.2.1, w1 ≥ 0 and w2 > −∞. Since Eu(0) < $w2 , by (E ) we get F u(t) ≥ w1 − Eu(0) for all t ∈ I, so that w2 ≥ w1 − Eu(0). Thus (1 + $)w2 > w1 ≥ 0 and so w2 > 0, being $ > −1. 2

90

4.3


Global Non-continuation Results

This section is devoted to the main general result for (4.1.1), where the damping Q is possibly nonlinear in u and v, so that on f we assume, beyond (F1 ) and (F2 ), a condition which is stronger than (F3 ), namely (F4 ) There exists a function q ∈ C+ (Ω) verifying (4.2.2) with the property that for all F > 0 and φ ∈ X for which inf t∈I F φ(t) ≥ F, there exist c1 = c1 (F, φ) > 0 and ε0 = ε0 (F, φ) > 0 such that F φ(t) ≤ c1 ρq(·) (φ(t, ·)) for all t ∈ I,

(i)

and for all ε ∈ (0, ε0 ] there exists c2 = c2 (F, φ, ε) > 0 such that (ii) hf (t, ·, φ(t, ·)), φ(t, ·)i−(q− −ε)F φ(t) ≥ c2 ρq(·) (φ(t, ·)) for all t ∈ I. Note that (F4 )–(ii) implies hf (t, ·, φ(t, ·)), φ(t, ·)i − (q− − ε)F φ(t) ≥ 0, so that letting ε → 0 we get (F3 ). In this section, we assume (F1 ), (F2 ) and (F4 ) on f . In order to prove Theorem 4.3.1, that is non-existence of global solutions for (4.1.1) in the entire R+ 0 × Ω, when the initial energy is appropriately bounded above, we also require the following condition on Q. (Q) Along every global solution u ∈ X of (4.1.1), there exist t ≥ 0, q1 > 0, m > 1, κ ≥ 0, ℘ > 1 with m+κ ≤ ℘ < q− , non-negative functions 1,1 δ1 , δ2 ∈ L∞ loc (J) and positive functions ψ, k ∈ Wloc (J), J = [t, ∞), with k 0 ≥ 0, such that for all t ∈ J 0 0 κ/m Q(t) ≤ q1 δ1 (t)1/m ku(t, ·)kq(·) Du(t)1/m + δ2 (t)1/℘ Du(t)1/℘ ku(t, ·)kq(·) , 1/(m−1)

δ1

1/(℘−1)

+ δ2

≤ k/ψ, Z

ψ 0 (t) = o(ψ(t)) as t → ∞, (4.3.1)

∞

Ψ(t)dt = ∞,

(4.3.2)

with Q(t) =hQ(t, ·, u(t, ·), ut (t, ·)),u(t, ·)i, Ψ(t) = ψ(t)max{k(t),ψ(t)}−(1+θ) , θ ∈ (0, θ0 ] and q− − 2 r 1 1 , , r= − ∈ (0, 1), (4.3.3) θ0 = min q− + 2 1 − r ℘ q− is an appropriate exponent.

91

Condition (Q) is a restriction only for large t. Whenever ψ(t) ≡ Const. > 0 in J, assumption (4.3.2) is equivalent to Z ∞ k(t)−(1+θ) dt = ∞, see Section 4.4 for further details, as well as for other specific examples of nonlinear dampings Q verifying (Q). When L = 1 in [15] the special case δ2 ≡ 0 and q ≡ Const. was assumed on Q, see also [11, 12] for problems under dynamic boundary conditions. For simplicity of notation let us introduce the negative numbers α1 = 1 +

κ q− − , m m

α2 = 1 −

q− . ℘

(4.3.4)

Actually, α1 ≤ α2 < 0, since (℘ − m)q− ≥ κq− ≥ κ℘, being κ ≥ 0, and ℘ < q− in (Q). Theorem 4.3.1. There are no global solutions u of (4.1.1) such that Eu(0) < E2 ,

(4.3.5)

where E2 is given in (4.2.3). Proof. We assume by contradiction that there exists a global solution u of (4.1.1) in R+ 0 × Ω, satisfying (4.3.5) as in the statement. Take φ = u in the Distribution Identity (A), thus d hu(t, ·), ut (t, ·)i =kut (t, ·)k22 − hAu(t, ·), u(t, ·)i dt + hf (t, ·, u(t, ·)), u(t, ·)i − hQ(t, ·, u(t, ·), ut (t, ·)), u(t, ·)i.

(4.3.6)

By Proposition 4.2.2–(i) and condition (4.3.5), it results that w2 > 0. We define for all t ∈ R+ 0 , the function Z t H (t) = H0 + Du(τ )dτ, (4.3.7) 0

where H0 is any positive number in the interval (0, E2 − Eu(0)). Of course H is well defined by (B)–(i) and non-decreasing, being D ≥ 0.

92


Let ε0 = ε0 (w2 , u) > 0 be the number corresponding to F = w2 and φ = u in (F4 ). Without loss of generality, we take ε0 > 0 so small that

ε0 w2 ≤ (q− − γp+ )w2 − γp+ (H0 + Eu(0)) = γp+ (E2 − H0 − Eu(0)), (4.3.8) which is possible by (4.3.5) and (4.3.7). We remark that (4.3.8) forces ε0 ≤ q− − γp+ . Fix ε ∈ (0, ε0 ]. By (4.2.1) and (F4 )–(ii)

hAu(t, ·), u(t, ·)i−hf (t, ·, u(t, ·)), u(t, ·)i ≤ γp+ A u(t) − (q− − ε)F u(t) − c2 ρq(·) (u(t, ·)).

Moreover, A u(t) ≤ Eu(t) + F u(t) and H (t) ≤ H0 + Eu(0) − Eu(t) by (E ) and (B)–(ii), so that

γp+ A u(t)−(q− − ε)F u(t) ≤ γp+ Eu(t) − (q− − ε − γp+ )F u(t) ≤ γp+ (H0 + Eu(0)) + [ε − (q− − γp+ )]w2 − γp+ H (t) ≤ −γp+ H (t),

by (4.3.8). Hence, combining these facts together with (4.3.6), and recalling the position Q(t) = hQ(t, ·, u(t, ·), ut (t, ·)), u(t, ·)i, we get

d hu(t, ·), ut (t, ·)i ≥ kut (t, ·)k22 +γp+ H (t)+c2 ρq(·) (u(t, ·))−Q(t). (4.3.9) dt

Let us estimate the main dynamic quantity Q(t) using (Q). By (4.3.1),

93

for all t ∈ J, 1/m0 1+κ/m δ1 (t)1/(m−1)Du(t) ku(t, ·)kq(·) o 1/℘0 + δ2 (t)1/(℘−1) Du(t) ku(t, ·)kq(·) ( h i 1/m0 q− /m 1 = q1 δ1 (t)1/(m−1)Du(t) ku(t, ·)kq(·) ku(t, ·)kαq(·)

Q(t) ≤ q1

+

n

h

) i 0 1/℘ q /℘ − 2 δ2 (t)1/(℘−1) Du(t) ku(t, ·)kq(·) ku(t, ·)kαq(·)

("

# 1/(m−1) 2 ` q− 1 ≤ q1 δ1 (t) Du(t) + ku(t, ·)kq(·) ku(t, ·)kαq(·) ` 2 " ) # 1/(℘−1) 2 ` q− 2 + , δ2 (t) Du(t) + ku(t, ·)kq(·) ku(t, ·)kαq(·) ` 2 (4.3.10) where in the last step we have applied Young’s inequality, with ` > 0 to be fixed later. By (F4 )–(i), in correspondence to F = w2 > 0 and φ = u ∈ X, there exists c1 = c1 (w2 , u) > 0 such that (F4 )–(i) is valid along u. In particular, putting c˜1 = min{(w2 /c1 )1/q− , (w2 /c1 )1/q+ } > 0, for all t ∈ R+ 0 ku(t, ·)kq(·) ≥ c˜1 > 0. By (4.3.4), being α1 ≤ α2 < 0, we obtain n Q(t) ≤ q2 `˜ δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) Du(t) o q− 2 , +`ku(t, ·)kq(·) ku(t, ·)kαq(·)

(4.3.11)

(4.3.12)

0 0 where q2 = q1 max{1, c˜1α1 −α2 } and `˜ = max{(2/`)m /m , (2/`)℘ /℘ }. By (4.3.7), (B)–(ii), (E ) and the definition of w2 , it follows that

H (t) ≤ H0 + Eu(0) − Eu(t) < E2 + F u(t) q− q− ≤ − 1 F u(t) + F u(t) = F u(t) γp+ γp+

(4.3.13)

94


for all t ∈ R+ 0 . Now, by (F4 )–(i) and (1.1.4), if ku(t, ·)kq(·) ≥ 1, then q+ . On the other hand, if ku(t, ·)kq(·) ≤ 1, then F u(t) ≤ c1 ku(t, ·)kq(·) q− w2 ≤ c1 ku(t, ·)kq(·) by (F4 )–(i), the definition of w2 and (1.1.4). Hence ku(t, ·)kq(·) ≥ (w2 /c1 )1/q− > 0, so that + , F u(t) ≤ c1 ρq(·) (u(t, ·)) ≤ c1 (c1 /w2 )(q+ −q− )/q− ku(t, ·)kq(·)

q

again by (F4 )–(i). In conclusion, along the solution u, we have for all t ∈ R+ 0 q+ , (4.3.14) F u(t) ≤ C1 ku(t, ·)kq(·) with C1 = c1 max{1, (c1 /w2 )(q+ −q− )/q− }. Clearly r=−

α2 ∈ (0, 1), q−

so that, by (4.3.13) and (4.3.14), we get −rq

rq /q

2 = ku(t, ·)kq(·) − ≤ C1 − + [F u(t)]−rq− /q+ ku(t, ·)kαq(·) rq− /q+ C 1 q− ≤ [H (t)]−rq− /q+ . γp+

Therefore by (4.3.12) n Q(t) ≤c3 `˜ δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) Du(t) o q− +`ku(t, ·)kq(·) [H u(t)]−rq− /q+ for all t ∈ J, where c3 = q2 (C1 q− /γp+ )rq− /q+ . Put 1 1 r0 = min r, − , 2 q−

(4.3.15)

(4.3.16)

(4.3.17)

and note that θ0 in (4.3.3) can be expressed as θ0 = r0 /(1−r0 ). From now on, we take r = θq− /[(1 + θ)q+ ], so that r ∈ (0, r0 q− /q+ ]. Consequently, we have n ˜ r−rq− /q+ δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) Q(t) ≤ c3 `H 0 o (4.3.18) −rq /q q− · [H (t)]−r Du(t) + `H0 − + ku(t, ·)kq(·) ,

95

where we used the facts that H ≥ H0 and that 0 < r ≤ r0 q− /q+ ≤ rq− /q+ by (4.3.17). Since Du = H 0 , we see that (1 − r)H −r H 0 = [H 1−r ]0 . Hence it is convenient to introduce the function Z (t) = λk(t) [H (t)]1−r + ψ(t)hu, ut i, 1,1 (J) by where λ > 0 is a constant to be fixed later. Clearly Z ∈ Wloc Corollary 8.10 of [23] and so, a.e. in J,

Z 0 = λk(1 − r)H −r H 0 + λk 0 H 1−r + ψ

d hu, ut i + ψ 0 hu, ut i. dt

By (4.3.9) and (4.3.18), it results that a.e. in J Z 0 ≥ λk(1 − r)H −r H 0 + λk 0 H 1−r + ψ 0 hu, ut i + ψ kut k22 + γp+ H (t) + c2 ρq(·) (u) − Q(t) n h i o ˜ r−rq− /q+ δ 1/(m−1) + δ 1/(℘−1) ψ H −r H 0 ≥ λk(1 − r) − c3 `H 0 1 2 + γp+ ψH (t) + λk 0 H 1−r + ψ 0 hu, ut i n o −rq /q q− + ψ kut k22 + c2 ρq(·) (u) − c3 `H0 − + kukq(·) . q

− If ku(t, ·)kq(·) ≥ 1, then by (1.1.4) ρq(·) (u(t, ·)) ≥ ku(t, ·)kq(·) . Similarly, if q+ q+ −q− q− ku(t, ·)kq(·) ≤ 1, then ρq(·) (u(t, ·)) ≥ ku(t, ·)kq(·) ≥ c˜1 ku(t, ·)kq(·) , by (1.1.4) and (4.3.11). Hence, in both cases,

q −q−

ρq(·) (u(t, ·)) ≥ min{1, c˜1 +

q

− }ku(t, ·)kq(·) .

Thus, by (4.3.1)2 and the fact that λk 0 H 1−r ≥ 0, we find i h ˜ r−rq− /q+ H −r H 0 + ψ 0 hu, ut i + γp+ ψH (t) Z 0 ≥ k λ(1 − r) − c3 `H 0 n o −rq /q q− + ψ kut k22 + c˜2 − c3 `H0 − + kukq(·) , q −q

a.a. in J, where c˜2 = c2 min{1, c˜1 + − }. Next, by the Cauchy and Young inequalities, and the definition of X, we have |hu(t, ·), ut (t, ·)i| ≤ kut (t, ·)k2 ku(t, ·)k2 ≤ kut (t, ·)k22 + ku(t, ·)k22 .

96


The embedding Lq(·) (Ω) ,→ L2 (Ω) is continuous by (4.2.2) and Theorem 1.1.1, so that ku(t, ·)k2 ≤ (1 + |Ω|)ku(t, ·)kq(·) , and by (4.3.11), this gives q

− , ku(t, ·)k22 ≤ (1 + |Ω|)2 ku(t, ·)k2q(·) ≤ c4 ku(t, ·)kq(·)

2−q−

where c4 = (1 + |Ω|)2 c˜1

(4.3.19)

, being q− > 2. Hence q

− , |hu(t, ·), ut (t, ·)i| ≤ kut (t, ·)k22 + c4 ku(t, ·)kq(·)

which, inserted in the preceding estimate of Z 0 , yields n o r−rq− /q+ 0 ˜ Z ≥k λ(1 − r) − c3 `H0 H −r H 0 + γp+ ψH (t) n o −rq /q q− . + ψ {1 − |ψ 0 /ψ|} kut k22 + ψ c˜2 − c4 |ψ 0 /ψ| − c3 `H0 − + kukq(·) There is T1 ∈ J such that 2|ψ 0 /ψ| ≤ min{1, c˜2 /c4 } in J1 = [T1 , ∞), since ψ 0 (t) = o(ψ(t)) as t → ∞. Then we take ` > 0 so small that 4c3 ` ≤ rq /q ˜ r−rq− /q+ /(1 − r), 1} c˜2 H0 − + and λ > 0 so large that λ ≥ max{c3 `H 0 and Z (T1 ) > 0, being H (T1 )1−r k(T1 ) > 0. In conclusion, we have shown that for a.a. t ∈ J1 n o q− Z 0 (t) ≥ c5 ψ(t) H (t) + kut (t, ·)k22 + ku(t, ·)kq(·) ≥ 0, (4.3.20) where 2c5 = min{˜ c2 /2, 1}. Therefore, Z (t) ≥ Z (T1 ) > 0 for all t ∈ J1 . On the other hand, from the definition of Z , we obtain α Z α ≤ λkH 1/α + ψ|hut , ui| (4.3.21) ≤ 2α−1 {(λk)α H + ψ α kut kα2 kukα2 } , where α = q− /(q− − rq+ ) and α ∈ (1, 2) by (4.3.17) and the choice of r. Put ν = 2/α, so that ν > 1. Hence, from (4.3.21) and Young’s inequality, for all t ∈ J1 n o α α α−1 αν αν 0 Z (t) ≤ 2 [max{λk(t), ψ(t)}] H (t) + kut (t, ·)k2 + ku(t, ·)k2 . Furthermore, 1 ν−1 1 1 1 1 = = − = −r ≥ 0 αν αν α 2 2 q−

97

by (4.3.17), and so αν 0 ≤ q− . Thus, using the argument produced in (4.3.19), we get 0

0

0

q

− ≤ (1 + |Ω|)αν ku(t, ·)kαν ku(t, ·)kαν q(·) ≤ c6 ku(t, ·)kq(·) , 2 0

q −αν 0

where c6 = (1 + |Ω|)αν /˜ c1− (F4 )–(i), it follows

(4.3.22)

. Therefore, by (4.3.22), (4.3.13) and

o n q− 2 Z (t) ≤ c7 max{λk(t), ψ(t)} H (t)+kut (t, ·)k2 +ku(t, ·)kq(·) , (4.3.23) α

α

where c7 = 2α−1 max{c6 , 1}. Combining this with (4.3.20) and λ ≥ 1, we obtain Z −α Z 0 ≥ c8 Ψ a.e. in J1 , (4.3.24) where c8 = c5 /c7 λα . Finally, since α = 1 + θ and r = θq− /(1 + θ)q+ , by (4.3.2) we see that Z cannot be global. Indeed, fix t ∈ J1 and integrate both sides of (4.3.24) in (T1 , t). By (4.3.2) it results Z t 1−α 1−α Z (t) ≤ Z (T1 ) − c8 (α − 1) Ψ(τ )dτ → −∞, T1

as t → ∞. This contradiction shows that u cannot be global in time. 2 As it is apparent from the proof of Theorem 4.3.1, when Eu(0) ≤ 0, so that clearly Eu(0) < E2 , then by (4.3.8) we can take ε = ε0 = q− − max{γp+ , 2} and condition (F4 ), when p ≡ 2, somehow reduces to the pioneering assumption of Levine and Serrin in [74]. Furthermore, in this case the choice of H0 = E2 is optimal for (4.3.18) and consequently for (4.3.20). In [74] and later in [104]–[107], assumptions (F1 )–(F2 ) and (F4 ) are required in a stronger form and the main geometry structure there implies in particular Eu(0) < E1 . (4.3.25) Note that if (4.3.25) holds, then w2 ≥ w1 − Eu(0) > w1 − E1 = max{γp+ , 2}w1 /q by (E ) and consequently E2 > E1 by (4.2.3). From Proposition 4.2.2–(i), it follows that if u is a solution of (4.1.1) such that w2 ≤ 0, then Eu(0) ≥ E2 . Hence, being w1 ≤ Eu(0) + w2 ≤ Eu(0) by (E ), also the case (4.3.25) can never occur, since E1 < w1 by (4.2.3).

98


4.4

Special Nonlinear External Damping

Recall that Q is continuous and satisfies inequality (4.1.2). Moreover we assume that there exists t >> 1 such that Q(t, x, u, v) = d1 (t, x)|u|κ |v|m−2 v + d2 (t, x, u)|v|℘−2 v, m > 1, κ ≥ 0, ℘ > 1, with m + κ ≤ ℘ < q− ,

(4.4.1)

for all (t, x, u, v) ∈ J × Ω × Rd × Rd , where J = [t, ∞), the functions + ∞ d ℘1 d1 ∈ C(R+ 0 → L (Ω)) and d2 ∈ C(R0 → L (Ω × R )) are non-negative and the constant ℘1 = q− /(q− − κ − m). Define δ1 (t) = kd1 (t, ·)k℘1 and δ2 (t) = |Ω|(q− −℘)/q− sup(x,ξ)∈Ω×Rd d2 (t, x, ξ). For the next proposition it is enough to assume on the nonlinear driving term f only conditions (F1 ) and (F2 ). Proposition 4.4.1. Let Q be of the type given in (4.4.1). Then, along every global solution u of (4.1.1) n 0 κ/m |Q(t)| ≤ (1 + |Ω|)1+κ/m δ1 (t)1/m ku(t, ·)kq(·) Du(t)1/m o 1/℘ 1/℘0 +δ2 (t) Du(t) ku(t, ·)kq(·)

(4.4.2)

for all t ∈ J. Moreover, if δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) ≤ K(1 + t)s/(m−1)

for all t ∈ J,

(4.4.3)

with K ≥ 1 and 0 ≤ s ≤ m − 1, then (Q) holds. Proof. Let u be a global solution of (4.1.1). By (4.4.1), for all (t, x) ∈ J ×Ω |Q(t, x, u(t, x), ut (t, x))| ≤ (d1 (t, x)|u(t, x)|κ )1/m(Q(t, x, u(t, x), ut (t, x)), ut (t, x))1/m 0

+ d2 (t, x, u(t, x))1/℘ (Q(t, x, u(t, x), ut (t, x)), ut (t, x))1/℘ .

0

99

So that, by Hölder’s inequality, we get in J kQ(t, ·,u(t, ·), ut (t, ·))k(q− )0 0

≤ k(d1 (t, x)|u|κ )1/m (Q(t, x, u, ut ), ut )1/m k(q− )0 0

+ kd2 (t, x, u)1/℘ (Q(t, x, u, ut ), ut )1/℘ k(q− )0 − −m Z qmq q− − 1 ≤ d1 (t, x)|u(t, x)|κ q− −m dx hQ(t, ·, u, ut ), ut (t, ·)i m0 Ω

+

|Ω|

!1/℘

q− −℘ q−

0

hQ(t, ·, u, ut ), ut (t, ·)i1/℘ .

d2 (t, x, ξ)

sup (x,ξ)∈Ω×Rd

On the other hand, applying once again Hölder’s inequality, we find that Z

κ

d1 (t, x)|u(t, x)|

q q−−m −

Z dx ≤

q− q− −m−κ

d1

q κ−m q−q −m−κ Z − −m − q− dx |u| dx .

Ω

Ω

Ω

Hence, combining the last two inequalities, we have by (F2 ), kQ(t, ·, u(t, ·), ut (t, ·))k(q− )0 κ/m 1/m ≤ kd1 (t, ·)k1/m ℘1 ku(t, ·)kq− hQ(t, ·, u, ut ), ut (t, ·)i

0

0

+ δ2 (t)1/℘ · hQ(t, ·, u, ut ), ut (t, ·)i1/℘ 0

0

1/m ≤ δ1 (t)1/m ku(t, ·)kκ/m +δ2 (t)1/℘ Du(t)1/℘ , q− Du(t) + ∞ being δ1 and δ2 as above of class C(R+ 0 ), so that δ1 , δ2 ∈ Lloc (R0 ). By Theorem 1.1.1, the embedding Lq(·) (Ω) ,→ Lq− (Ω) is continuous, hence by Hölder’s inequality we get

|Q(t)| ≤ kQ(t, ·, u, ut )k(q− )0 ku(t, ·)kq− ≤ (1 + |Ω|)kQ(t, ·, u, ut )k(q− )0 ku(t, ·)kq(·) n 0 κ/m ≤ (1 + |Ω|)1+κ/m δ1 (t)1/m ku(t, ·)kq(·) Du(t)1/m o 0 +δ2 (t)1/℘ Du(t)1/℘ ku(t, ·)kq(·) , that is (4.3.1)1 holds, with q1 = (1 + |Ω|)1+κ/m . It remains to provide the functions ψ and k as in (Q) satisfying (4.3.1)2 as well as (4.3.2)–(4.3.3).

100


Case 0 ≤ s < m − 1. Take k(t) = K(1 + t)s/(m−1) and ψ(t) = 1 for all t ∈ J. Case s = m − 1. Take k(t) = K and ψ(t) = (1 + t)−1 , for all t ∈ J. 1,1 (J), k > 0, k 0 ≥ 0, ψ > 0, Hence, in both the situations ψ, k ∈ Wloc 0 ψ (t) = o(ψ(t)) as t → ∞, k ≥ ψ in J, being K ≥ 1, and (4.3.1)2 is verified in J. Moreover, with the notations of (4.3.2), for all t ∈ J we have

( (1 + t)−s(1+θ)/(m−1) , if 0 ≤ s < m − 1, −(1+θ) Ψ(t) = K (1 + t)−1 , if s = m − 1.

(4.4.4)

If s = 0 or s = m − 1, then (4.3.2) holds taking any θ ∈ (0, θ0 ], with θ0 as in (4.3.3), and the value θ = θ0 is optimal. If 0 < s < m − 1, then (4.3.2) holds, provided that θ > 0 is so small that θ ≤ min{θ0 , (m − 1 − s)/s}. 2

4.5

Energy Estimates

In this section we present some energy estimates which are related only to A and F . To this aim we only need conditions (F1 )–(F3 ) and (F4 )– (i) on f , and we introduce a prototype function satisfying them. More precisely, we assume

M (τ ) = a+bγτ γ−1 ,

a, b ≥ 0,

a+b > 0,

( > 1, γ = 1,

if b > 0, (4.5.1) if b = 0,

f (t, x, u) = g(t, x)|u|σ(x)−2 u + c(x)|u|q(x)−2 u,

(4.5.2)

where σ, q ∈ C+ (Ω), c ∈ L∞ (Ω) is a non-negative function such that c∞ = kck∞ > 0, g ∈ C(R+ 0 × Ω) is differentiable with respect to t, with

101

gt ∈ C(R+ 0 × Ω); moreover σ+ ≤ q− , max{2, γp+ } < q− , ( ≤ p∗L (x), if Lp+ < n, q(x) for all x ∈ Ω, < ∞, if Lp− ≥ n where p∗L is defined in (1.1.7) 1 0 ≤ −g(t, x), gt (t, x) ≤ h(x) in R+ 0 × Ω, for some h ∈ L (Ω),

where

(4.5.3)

g(t, ·) ∈ Lη(·) (Ω) in R+ 0, ( q(x)/(q(x) − σ(x)), if σ+ < q− , η(x) = ∞, if σ+ = q− .

The special functions M and f in (4.5.1) and (4.5.2) verify the basic conditions (M ) and (4.1.3). Actually, (4.5.2) takes inspiration from the Example (3.2) of [97], see also [11, 12, 15]. The negative term of (f (t, x, u), u), governed by g, plays the role of a nonlinear perturbation acting against the blow up. Proposition 4.5.1. Conditions (F1 )–(F3 ) and (F4 )–(i) are satisfied, with c1 = c∞ /q− . Proof. Fix φ ∈ X. Clearly, by (4.5.2) |(f (t, x, φ(t, x)), φ(t, x))| ≤ −g(t, x)|φ(t, x)|σ(x) + c∞ |φ(t, x)|q(x) , so that (f (t, x, φ(t, x)), φ(t, x)) ∈ L1 (Ω) for all t ∈ I and by (4.5.3) hf (t, ·, φ(t, ·)), φ(t, ·)i ∈ L1loc (I). Analogously, being F (t, x, φ) = g(t, x)

|φ(t, x)|σ(x) |φ(t, x)|q(x) + c(x) , σ(x) q(x)

then also F (t, x, φ(t, x)) ∈ L1 (Ω) for all t ∈ I. Hence (F1 ) holds. Furthermore, F φ(t) = F (t, φ) Z |φ(t, x)|σ(x) |φ(t, x)|q(x) = g(t, x) + c(x) dx. σ(x) q(x) Ω

(4.5.4)

102

Chapter 4. Global Non-continuation L,p(·)

The same expression holds for F (t, w) when (t, w) ∈ R+ 0 × W0 Thus, differentiation under the integral sign gives Z |w(x)|σ(x) Ft (t, w) = gt (t, x) dx. σ(x) Ω

(Ω).

L,p(·)

Hence Ft ≥ 0 in R+ (Ω) by (4.5.3), and so (F2 ) is fulfilled. 0 × W0 Moreover, for all F ≥ 0 and t ∈ I hf (t, x, φ(t, x)), φ(t, x)i−q− F φ(t) Z q− ≥ 1− g(t, x)|u(t, x)|σ(x) dx σ+ Ω Z q− + 1− c(x)|u(t, x)|q(x) dx ≥ 0 q− Ω Z c∞ |φ(t, x)|q(x) dx ≤ ρq(·) (φ(t, ·)), (4.5.5) F φ(t) ≤ c(x) q(x) q− Ω by (4.5.4), being σ+ ≤ q− and g ≤ 0. In other words, (F3 ) and (F4 )–(i) hold for all F ≥ 0, as stated. 2 We now proceed with a qualitative analysis based on the geometric features of the models under consideration, which is significant for the evolution problems, see [11]–[15],[19, 29, 72, 74, 83] and also [88]–[109]. Put s = b if b > 0 or s = a if b = 0. From now on in the section, we fix a solution u ∈ X of (4.1.1) and, in correspondence to u, put υ(t) = ku(t, ·)kq(·) .

Proposition 4.5.2. The following inequality holds for all t ∈ I Eu(t) ≥ ϕ(υ(t)), (4.5.6) sΛ c∞ ϕ(υ(t)) = γ min{υ(t)γp+ , υ(t)γp− } − max{υ(t)q+ , υ(t)q− }, p+ q− γ

where Λ = min{(Sq0 + )p+ , (Sq0 + )p− }, with Sq0 + = 1/κL Sq+ and κL and Sq+ are the constants defined in Proposition 1.2.2 and Theorem 1.1.8, respectively.

103

Proof. By (E ) we have Eu(t) ≥ A u(t) − F u(t) for each t ∈ I. From Proposition 4.5.1 and (1.1.4), F u(t) ≤

c∞ c∞ ρq(·) (φ(t, ·)) ≤ max{υ(t)q+ , υ(t)q− }. q− q−

(4.5.7)

By (A ), (4.5.1) and (1.1.4), we have A u(t) ≥ M (IL (u(t))) = aIL (u(t)) + bIL (u(t))γ a b ≥ ρp(·) (DL u(t, ·)) + γ ρp(·) (DL u(t, ·))γ p+ p+ a p− p+ } ≥ min{kDL u(t, ·)kp(·) , kDL u(t, ·)kp(·) p+ b γp γp + γ min{kDL u(t, ·)kp(·)+ , kDL u(t, ·)kp(·)− }. p+ Now, by Theorem (1.1.8) with h = q and hypothesis (4.5.3), the emL,p(·) bedding W0 (Ω) ,→ Lq(·) (Ω) is continuous, so that for an appropriate constant Sq+ = Sq+ (n, d, p, L, Ω) > 0, it results ku(t, ·)kq(·) ≤ Sq+ ku(t, ·)kW L,p(·) (Ω) . Hence, by Proposition 1.2.2, we get 0

kDL u(t, ·)kp(·) ≥

1 ku(t, ·)kW L,p(·) (Ω) ≥ Sq0 + ku(t, ·)kq(·) = Sq0 + υ(t). 0 κL

Therefore, a min{(Sq0 + υ(t))p+ , (Sq0 + υ(t))p− } p+ b + γ min{(Sq0 + υ(t))γp+ , (Sq0 + υ(t))γp− } p+ (4.5.8) bΛγ aΛ min{υ(t)p+ , υ(t)p− } + γ min{υ(t)γp+ , υ(t)γp− } ≥ p+ p+ γ sΛ ≥ γ min{υ(t)γp+ , υ(t)γp− }. p+

A u(t) ≥

Therefore, combining together (4.5.7) and (4.5.8), we conclude the proof. 2

104


From Lemma 4.5.2 we obtain Eu(t) ≥ ϕ(υ(t)) for all t ∈ I, where ϕ : R+ 0 → R is defined by ϕ(υ) = ϕ0 (υ) if 0 ≤ υ ≤ 1, while ϕ(υ) = ϕ1 (υ) if υ ≥ 1, with ϕ0 (υ) =

sΛγ γp+ c∞ q− − υ , υ pγ+ q−

ϕ1 (υ) =

It is easy to see that, if we take ( Λ ≤ min (Sq0 + )p+ , (Sq0 + )p− ,

sΛγ γp− c∞ q+ − υ . υ pγ+ q−

c∞ pγ−1 + γs

1/γ ) (4.5.9)

ϕ attains its maximum at 1/(q− −γp+ )

υ0 = a0

,

where a0 =

sγΛγ . c∞ pγ−1 +

(4.5.10)

The choice of Λ in (4.5.9) guarantees that υ0 ∈ (0, 1]. Moreover, ϕ1 takes 1/(q −γp ) its maximum at υ1 = a1 + − , where a1 = a0 p− q− /p+ q+ ≤ a0 ≤ 1. Hence ϕ is strictly decreasing for υ ≥ υ0 , with ϕ(υ) → −∞ as υ → ∞. Put γp γp+ sΛγ υ0 + ϕ(υ0 ) = 1 − w0 = E0 > 0, where w0 = > 0, q− pγ+ Σ0 = {(υ, E) ∈ R2 : υ > υ0 , E < E0 }. (4.5.11) The next two lemmas constitute the key tool for the subsequent applications, since they contain the principal information deriving from the geometry of the system. Lemma 4.5.3. If Eu(0) < E0 then υ0 ∈ / υ(I). Moreover, the following conditions are equivalent: (i) w1 ≥ w0 ; (ii) υ(I) ⊂ (υ0 , ∞); (iii) w2 > γp+ w0 /q− . Finally, if one of the conditions (i)–(iii) holds, then E0 ≤ E1 < E2 . In particular, if υ(0) > υ0 and Eu(0) < E0 , then (υ(t), Eu(t)) ∈ Σ0 for all t ∈ I, properties (i)–(iii) hold, E0 ≤ E1 < E2 and w2 > γp+ w1 /q− .

105

Proof. Assume that Eu(0) < E0 and suppose by contradiction that υ0 ∈ υ(I). It follows that there exists a sequence (tj )j in I such that υ(tj ) → υ0 as j → ∞. Now, by (B)–(ii) and (4.5.6) we have E0 > Eu(0) ≥ Eu(tj ) ≥ ϕ(υ(tj )), which provides E0 > E0 by the continuity of ϕ ◦ υ. This contradiction proves the claim. (i) ⇒ (ii). It is enough to prove that υ(I) ⊂ (υ0 , ∞), which immediately gives υ(I) ⊂ (υ0 , ∞), since υ0 ∈ / υ(I). By (E ) and (4.5.7) γp+ γp+ γp+ 1− A u(t) ≥ 1 − w1 ≥ 1 − w0 q− q− q− = E0 > Eu(0) ≥ Eu(t) c∞ ≥ A u(t) − F u(t) ≥ A u(t) − max{υ(t)q+ , υ(t)q− }, q− therefore A u(t) ≤

c∞ max{υ(t)q+ , υ(t)q− }. γp+

(4.5.12)

On the other hand, by (4.5.8) A u(t) ≥

sΛγ min{υ(t)γp+ , υ(t)γp− }. pγ+

(4.5.13)

Now, if t ∈ I is such that υ(t) > 1, automatically υ(t) > υ0 , being υ0 ≤ 1. If υ(t) ≤ 1 for some t ∈ I, then combining together (4.5.13) and (4.5.12) we get immediately that υ(t) > υ0 . Hence υ(t) > υ0 for all t ∈ I. (ii) ⇒ (iii). In this case υ(t) > υ0 for all t ∈ I. Hence, by (4.5.8) γp γp γp Au(t) > sΛγ min{υ0 + , υ0 − }/pγ+ = sΛγ υ0 + /pγ+ = w0 for all t ∈ I, so that F u(t) ≥ A u(t) − Eu(0) > w0 − Eu(0) > w0 − E0 = γp+ w0 /q− for all t ∈ I by (E ) and in turn w2 > γp+ w0 /q− . (iii) ⇒ (i). First we prove that υ(t) > υ0 for all t ∈ I. Indeed, if υ(t) > 1 we are done. If υ(t) ≤ 1, then by (4.5.7) we have F u(t) ≤ c∞ υ(t)q− /q− .

(4.5.14)

γp+ sγΛγ γp+ F u(t) ≥ w2 > w0 = υ0 . q− q− pγ−1 +

(4.5.15)

On the other hand,

106


Hence, by (4.5.14), (4.5.15) and (4.5.10) we get q−

υ(t)

sγΛγ γp+ q > = υ0− γ−1 υ0 c∞ p +

and the entailment is proved thanks to (4.5.8). Finally, if one of the conditions (i)–(iii) holds, then E0 ≤ E1 by (i). Furthermore, F u(t) ≥ w1 − Eu(0) > w1 − E1 = γp+ w1 /q− for all t ∈ I by (E ). Hence, w2 > γp+ w1 /q− and so E1 < E2 . In conclusion E0 ≤ E1 < E2 , as claimed. The last part of the lemma follows at once from the previous arguments. 2 In the case p ≡ Const. and q ≡ Const., Lemma 4.5.3 can be improved in the following form. Lemma 4.5.4. Let p ≡ Const. and q ≡ Const. If Eu(0) < E0 then υ0 ∈ / υ(I) and w1 6= w0 . Moreover, the following conditions are equivalent: (i) w1 > w0 ; (ii) υ(I) ⊂ (υ0 , ∞); (iii) w2 > γpw0 /q. Finally, if one of the conditions (i)–(iii) holds, then E0 < E1 < E2 . In particular, if υ(0) > υ0 and Eu(0) < E0 , then (υ(t), Eu(t)) ∈ Σ0 for all t ∈ I, properties (i)–(iii) hold, E0 < E1 < E2 and w2 > γpw1 /q. Proof. It is enough to prove that if Eu(0) < E0 , then w1 6= w0 . Clearly, υ(t) > υ0 for all t ∈ I, as shown in Lemma 4.5.3. Suppose by contradiction that w1 = w0 , then there exists a sequence (tj )j such that A u(tj ) → w1 = w0 as j → ∞. By (4.5.8) γ 1/γp p υ(t) ≤ A u(t) sΛγ for all t ∈ I. Hence, lim sup υ(tj ) ≤ j→∞

which is impossible.

pγ w1 sΛγ

1/γp

=

pγ w0 sΛγ

1/γp = υ0 , 2

107

In special reverse situations in which, considered T = ∞, the external force f is of restoring type, that is (f (t, x, u), u) ≥ 0 for all (t, x, u) ∈ d R+ 0 × Ω × R , it is possible to show that if υ(0) < υ0 and Eu(0) < E0 , then υ(t) < υ0 and Eu(t) < E0 for all t ∈ R+ 0 . In particular, any point (υ(t), Eu(t)) on the trajectory of a global solution u ∈ X must remain in the potential well, see, e.g., [96, Remark on page 45] for dissipative wave systems, reference [14] for the case L = 1, and [19] for abstract evolution equations and applications to the polyharmonic operators. In order to handle the delicate case Eu(0) = E0 , we introduce the following condition. (D) There exists t∗ ∈ (0, T ) such that one of the following properties holds: (i) φ ∈ X and hQ(t, ·, φ, φt ), φt i ≡ 0 in [0, t∗ ] implies either φ(t, ·) ≡ 0 or φt (t, ·) ≡ 0 in [0, t∗ ], and Q verifies for all (t, x, u, v) in R+ 0 × Ω × Rd × Rd Q(t, x, u, v) = d1 (t, x)|u|κ |v|m−2 v + d2 (t, x, u)|v|℘−2 v,

(4.5.16)

where d1 , d2 and all the exponents are as in (4.4.1); + (ii) there exists a positive function g0 : R+ 0 → R such that gt (t, x) ≥ g0 (t) for each (t, x) ∈ [0, t∗ ] × Ω.

Lemma 4.5.5. Assume (D). If υ(0) > υ0

and

Eu(0) = E0 ,

(4.5.17)

then υ(I) ⊂ (υ0 , ∞). Furthermore (υ(t), Eu(t)) ∈ Σ0 for all t ∈ (0, T ), that is υ(t) > υ0

and

Eu(t) < E0

for all t ∈ (0, T ).

(4.5.18)

In particular, w1 ≥ w0 , w2 ≥ γp+ w1 /q− and E0 ≤ E1 ≤ E2 . Proof. First we show that υ(t) 6= υ0 for all t ∈ I. Proceed by contradiction and suppose that there exists t0 ∈ I such that υ(t0 ) = υ0 . Then, by (4.5.6) and the assumption Eu(0) = E0 , it follows E0 = Eu(0) ≥ Eu(t0 ) ≥ ϕ(υ(t0 )) = E0 .

108


Hence E0 = Eu(t0 ), so that, by (B)–(ii) and the fact that Du ≥ 0, we get Z t0 Du(t)dt ≤ Eu(0) = E0 . E0 = Eu(t0 ) ≤ Eu(0) − 0

R t0

Therefore 0 Du(t)dt = 0 and in turn Du ≡ 0 in [0, t0 ]. Consequently, by the definition of Du, we have that hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i = 0 and Ft u(t) = 0 for all t ∈ [0, s0 ], where s0 = min{t∗ , t0 } and t∗ is the number given in (D). Let us now distinguish two cases. Case (D)–(i). Since hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i = 0 for all t ∈ [0, s0 ], we get that either u(t, ·) = 0 or ut (t, ·) = 0 for all t ∈ [0, s0 ]. The first event cannot occur since υ(0) = ku(0, ·)kq(·) > υ0 > 0 by assumption. In the latter, u is clearly constant with respect to t in [0, s0 ], and so u(t, x) = u(0, x) for each t ∈ [0, s0 ]. Taking φ(t, x) = u(0, x) in the Distribution Identity (A), for each t ∈ [0, s0 ] we have thAu(0, ·), u(0, ·)i = Rt hf (τ, ·, u(0, ·)), u(0, ·)idτ , since (4.4.2) holds in R+ 0 × Ω by (4.5.16), 0 Du ≡ 0 in [0, s0 ] and so hQ(t, ·, u(0, ·), 0), u(0, ·)i = 0. Thus hAu(0, ·), u(0, ·)i = hf (t, ·, u(0, ·)), u(0, ·)i for each t ∈ [0, s0 ], and so hAu(0, ·), u(0, ·)i = hf (0, ·, u(0, ·)), u(0, ·)i. Now γp+ A u(0) ≥ q− F u(0) by (4.2.1) and (F3 ). On the other hand, E0 = Eu(0) = A u(0) − F u(0) by (E ), since ut (0, ·) = 0. Hence, γp+ E0 ≥ 1 − A u(0). q− By (4.5.8) and (4.5.17) we have A u(0) > w0 > 0, and so γp+ γp+ E0 ≥ 1 − A u(0) > 1 − w0 = E0 , q− q− which is an obvious contradiction. Case (D)–(ii). We have Z |u(t, x)|σ(x) g0 (t) gt (t, x) dx ≥ ρσ(·) (u(t, ·)) ≥ 0 0 = Ft u(t) = σ(x) σ+ Ω for each t ∈ [0, s0 ]. Therefore, ρσ(·) (u(t, ·)) ≡ 0 and so u ≡ 0 in [0, s0 ] × Ω. But this occurrence is impossible, since υ(0) = ku(0, ·)kq(·) > υ0 > 0, so that we reach a contradiction.

109

Consequently, υ(t) > υ0 for all t ∈ I, by the continuity of t 7→ υ(t), being υ(0) > υ0 by assumption. Now Eu(t) ≤ E0 for all t ∈ I Rand there are no points t0 ∈ (0, T ) t such that Eu(t0 ) = E0 . Otherwise 0 0 Du(t)dt = 0 and assumption (D) would provide a contradiction, as shown above. Hence Eu(t) < E0 for all t ∈ (0, T ) and this concludes the proof of (4.5.18). By (4.5.8), we have for all t ∈ I A u(t) ≥

sΛγ sΛγ γp+ γp+ γp− = w0 , min{υ(t) , υ(t) } > υ pγ+ pγ+ 0

(4.5.19)

therefore w1 ≥ w0 and so E1 ≥ E0 . Furthermore, F u(t) ≥ w1 − Eu(0) = w1 − E0 ≥ w1 − E1 = γp+ w1 /q− for all t ∈ I by (E ), so that E1 ≤ E2 . Hence, E0 ≤ E1 ≤ E2 , as required. 2 In the case p ≡ Const. and q ≡ Const., Lemma 4.5.5 holds in the following stronger form. Lemma 4.5.6. Let p ≡ Const., q ≡ Const. and assume (D). If υ(0) > υ0

and

Eu(0) = E0 ,

then υ(I) ⊂ (υ0 , ∞). Furthermore (υ(t), Eu(t)) ∈ Σ0 for all t ∈ (0, T ). In particular, w1 > w0 , w2 > γpw1 /q and E0 < E1 < E2 . Proof. Clearly, also in this case Lemma 4.5.5 holds. Hence, υ(t) > υ0 for all t ∈ I, w1 ≥ w0 and it remains to show that w1 6= w0 . Otherwise, there exists a minimizing sequence (tj )j ⊂ I such that limj→∞ A u(tj ) = w0 and (tj )j cannot be contained in an interval [0, τ0 ], with τ0 < T , being by (4.5.19) A u > w0 for all t ∈ I. Hence, (tj )j admits a subsequence, still denoted by (tj )j , converging to T as j → ∞. Therefore lim inf t→T − A u(t) ≤ w0 and this forces lim inf t→T − A u(t) = w0 , being w0 = inf t∈I A u(t). Put now υ = inf t∈I υ(t), so that A u(t) ≥ sΛγ υ(t)γp /pγ ≥ sΛγ υ γp /pγ for all t ∈ I by (4.5.8). Consequently, w0 ≥ sΛγ υ γp /pγ , which yields υ0 ≥ υ. On the other hand, υ0 ≥ υ0 being υ(t) > υ0 for all t ∈ I by (4.5.18) and so υ0 = υ0 . Arguing as for A , we obtain that lim inf t→T − υ(t) = υ0 . Finally, w0 − c∞ υ(t)q /q < A u(t) − F u(t) ≤ Eu(t) ≤ E0 by (E ), (4.5.5) and (B)–(ii), so that lim supt→T − Eu(t) = E0 and so, by (B)–(ii), (4.1.2) and (F2 ), it follows

110


Rt RT limt→T − 0 Du(τ )dτ = 0 Du(τ )dτ = 0, so that Du ≡ 0 in I. This is impossible as shown in the proof of Lemma (4.5.5). In conclusion, w1 > w0 and so E0 < E1 . Furthermore, F u(t) ≥ w1 − Eu(0) = w1 − E0 > w1 − E1 = γpw1 /q for all t ∈ I by (E ), so that w2 > γpw1 /q and E1 < E2 . Hence, E0 < E1 < E2 , as required. 2 In the final part of this section we present an application of Theorem 4.3.1. To this aim, we require a further assumption on f , in order to get the validity of (F4 )–(ii), that is we assume σ+ < q−

and

c = ess inf Ω c(x) > 0,

(4.5.20)

so that (F4 )–(ii) is fulfilled with ε0 ∈ (0, q− − σ+ ] and c2 = cε/q− for all ε ∈ (0, ε0 ]. To prove the validity of (F4 )–(ii), fix φ ∈ X and ε0 ∈ (0, q− − σ+ ]. Then by (4.5.2)–(4.5.4), hf (t, ·, φ(t, ·)), φ(t, ·)i − (q− − ε)F φ(t) Z q− − ε = 1− g(t, x)|φ(t, x)|σ(x) dx σ Ω Z + ε + c(x)|φ(t, x)|q(x) dx q− Ω cε ≥ ρq(·) (φ(t, ·)), q− as claimed. In the next theorem we assume (4.4.1), (4.4.3), (4.5.1)–(4.5.3) and (4.5.20), while N is as always, so that all the structural assumptions of Theorem 4.3.1 are fulfilled; cf. Propositions 4.4.1 and 4.5.1. From now on we shall use the following notation u0 = u(0, ·) and u1 = ut (0, ·). Theorem 4.5.7. There are no global solutions u ∈ X of (4.1.1) in R+ 0 × Ω, satisfying ku0 kq(·) > υ0 ,

Eu(0) < E0 ,

(4.5.21)

where E0 is defined in (4.5.11). If furthermore, condition (D) holds, then there are no global solutions u ∈ X of (4.1.1) in R+ 0 × Ω, satisfying (4.5.17), that is such that ku0 kq(·) > υ0 ,

Eu(0) = E0 .

111

Proof. Assume by contradiction that u ∈ X is a solution of (4.1.1) + in R+ 0 × Ω, verifying (4.5.21). By Lemma 4.5.3, υ(0) 6∈ υ(R0 ), so that, υ(R+ 0 ) ⊂ (υ0 , ∞), being υ(0) > υ0 . Hence w2 > γp+ w0 /q− again by Lemma 4.5.3–(iii). Thus Eu(0) < E0 < E2 and the contradiction follows at once by an application of Theorem 4.3.1. Assume now also condition (D). Proceeding again by contradiction, let u ∈ X be a global solution of (4.1.1). Then Eu(0) = E0 ≥ E2 , otherwise by Theorem 4.3.1 there are no global solutions of (4.1.1). Hence, γp+ q− 1− w0 ≥ − 1 w2 , q− γp+ that is

γp+ w0 ≥ w2 . q−

Moreover, by Lemma 4.5.5 we have that w2 ≥

γp+ γp+ w1 ≥ w0 . q− q−

Therefore, w2 = γp+ w0 /q− . This implies that F u(t) ≥ γp+ w0 /q− for all t ∈ R+ 0 and we assert that equality cannot occur at a finite time. Indeed, if there were t such that F u(t) = γp+ w0 /q− , then υ(t) > υ0 by Lemma 4.5.5 and so by (4.5.8) Eu(0) ≥ Eu(t) ≥ A u(t) − F u(t) ≥ >

sΛγ γp+ γp+ w0 , υ(t)γp− }− γ min{υ(t) p+ q−

sΛγ γp+ γp+ υ w0 = E 0 , − pγ+ 0 q−

which contradicts (4.5.17). Therefore, it remains to consider the case w2 = γp+ w0 /q− and F u(t) > γp+ w0 /q− . A continuity argument shows at once that lim inf F u(t) = w2 . t→∞

Now, by (E ), (B)–(ii) and Lemma 4.5.5 Eu(t) ≥ A u(t) − F u(t) ≥ w1 − F u(t) ≥ w0 − F u(t),

112


R∞ so that lim supt→∞ Eu(t) = E0 . Hence, 0 Du(τ )dτ = 0 and in particular Du ≡ 0 in R+ 0 . Consequently, by (F2 ) and (4.4.1), we obtain hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i = 0 and Ft u(t) = 0 for all t ∈ R+ 0. Now, if (D)–(i) holds, since hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i = 0 for all t ∈ [0, t∗ ], we get that either u(t, ·) = 0 or ut (t, ·) = 0 for all t ∈ [0, t∗ ]. The first case cannot occur since ku(0, ·)kq(·) = υ(0) > υ0 > 0. In the latter, u is clearly constant with respect to t in [0, t∗ ], and so u(t, x) = u(0, x) for each t ∈ [0, t∗ ]. Taking φ(t, x) = u(0, x) in the Distribution R t Identity (A), then for each t ∈ [0, t∗ ] we have thAu(0, ·), u(0, ·)i = hf (τ, ·, u(0, ·)), u(0, ·)idτ , since hQ(t, ·, u(0, ·), 0), u(0, ·)i = 0 by (4.4.2), 0 being Du = 0 in [0, t∗ ]. Therefore hAu(0, ·),u(0, ·)i = hf (t, ·, u(0, ·)),u(0, ·)i for each t ∈ [0, t∗ ], and so hA(u(0, ·)), u(0, ·)i = hf (0, ·, u(0, ·)), u(0, ·)i. Now γp+ A u(0) ≥ q− F u(0) by (4.2.1) and (F3 ). On the other hand, E0 = Eu(0) = A u(0) − F u(0) by (E ), since ut (0, ·) = 0. By (4.5.8) and Lemma 4.5.5 we have A u(0) ≥ and so

sγ Λγ sγ Λγ γp+ γp− γp+ } > υ(0) = w0 , min{υ(0) υ pγ+ pγ+ 0

E0 ≥

γp+ 1− q−

γp+ A u(0) > 1 − w0 = E 0 q−

by (4.5.11). This is an obvious contradiction. While, if (D)–(ii) holds, then Z |u(t, x)|σ(x) g0 (t) gt (t, x) 0 = Ft u(t) = dx ≥ ρσ(·) (u(t, ·)) ≥ 0 σ(x) σ+ Ω for each t ∈ [0, t∗ ]. Therefore ρσ(·) (u(t, ·)) ≡ 0 and in turn u ≡ 0 in [0, t∗ ]× Ω, by (1.1.4). Again, as already shown, this occurrence is impossible, since υ(0) > υ0 , so that we reach a contradiction. This contradiction shows the claim. 2 The first part of Theorem 4.5.7 extends an analogous result for polyharmonic wave equations, due to Vitillaro in Section 4.4 of [104], as well as Theorem 2.2 for polyharmonic Kirchhoff equations of [83]. Similarly, the second part of Theorem 4.5.7 extends in several directions a previous result of Vitillaro contained in Section 4.4 of [104].

113

Indeed, here we treat a wider class of solutions, as noted in the Introduction, and are able to consider also the case Q ≡ 0, thanks to condition (D)–(ii), which was not investigated in [104]. Finally, Theorem 4.5.7 extends also Theorem 3.1 of [105], where L = 1, M ≡ 1, N ≡ 0 and the damping is linear in v, and Theorem 2.1 of [29].

4.6

Lifespan Estimates

In this section we give some applications of Theorem 4.3.1 for (4.1.1). Even if the next result can be proved in the generality of the main Theorem 4.3.1, we prefer to take M and f verifying (4.5.1)–(4.5.3) and (4.5.20), and we suppose that N is as always in this chapter. Moreover, 1,1 (R+ Q is of the type given in (4.5.16) and there exists k ∈ Wloc 0 ), with 0 + k ≥ 0 in R , k0 = k(0) > 0, verifying 1/(m−1)

δ1

1/(℘−1)

+ δ2

≤k

in R+ 0,

(4.6.1)

and Z

∞

k(t)−(1+θ) dt = ∞,

(4.6.2)

for some θ ∈ (0, θ0 ], where θ0 > 0 is defined in (4.3.3). Put ( 1/q− 1/q+ ) γp+ w0 γp+ w0 C1 = min 1, , c∞ c∞ ( rq− /q+ r) (4.6.3) c c ∞ ∞ C2 = (1 + |Ω|)1+κ/m C1α1 −α2 max 1, , γp+ γp+ w0 where c∞ and w0 are defined in (4.5.2) and (4.5.11), respectively, and α1 ≤ α2 < 0 are given in (4.3.4). Theorem 4.6.1. Let u ∈ X be a solution of (4.1.1) such that condition (4.5.21) holds, that is ku0 kq(·) > υ0 and Eu(0) < E0 . Consider the

114


numbers H0 = E0 − Eu(0) > 0, ε0 = min{q− − σ+ , q− − γp+ − q− E0 /w0 } > 0, ( ) 4qC2 1 1 θq− −rq− /q+ L = max 1, , r= − , r= , q+ −q− H0 ℘ q− (1 + θ)q+ cε0 C1 ( ) − 2hu , u i (1 + θ)q+ 1 0 0 1 r−rq /q λ = max C2 Lm /m H0 − + , , , 1−θq /[(1+θ)q − +] (1 + θ)q+ − θq− k0 k0 H0 1−θq /[(1+θ)q ]

− + Z0 = λk0 H0 + hu0 , u1 i, q q+ min{cε0 C1 /2q− , C1 − } K= θ . 2 [C1 (1 + |Ω|)]2(1+θ)/(1−θ)

(4.6.4) Then T ≤ T0 , where T0 is the unique positive number for which θ Z T0 λ λ −(1+θ) . (4.6.5) k(t) dt = θK Z0 0 Proof. Let u ∈ X be a solution of (4.1.1) as in the statement. By the first part of Lemma 4.5.3 we have that υ0 6∈ υ(I) so that υ(I) ⊂ (υ0 , ∞) by (4.5.21). Hence, by Lemma 4.5.3–(iii) we have E2 = (q− /γp+ −1)w2 > E0 > Eu(0). We follow essentially the argument of the proof of Theorem 4.3.1 with ψ ≡ 1. Indeed, the main structural assumptions (F1 )–(F2 ) and (F4 ) are verified by virtue of (4.5.1)–(4.5.3) and (4.5.20), see Proposition 4.5.1 and the comments to (4.5.20). In particular, we are able to take ε0 as in (4.6.4), ε = ε0 and c2 = cε0 /q− in (F4 ). Furthermore, (4.4.2) holds in I by (4.5.16), and so (4.3.1) is fulfilled, with ψ ≡ 1, by (4.6.1). Moreover, (4.6.2) is equivalent to (4.3.2), being ψ constant, as noted in Section 4.4. Define for all t ∈ I the function H as in (4.3.7), with H0 given in (4.6.4). Using the Distribution Identity (A), with φ = u, we get, as in the proof of Theorem 4.3.1, the inequality (4.3.9), that is d c ε0 ρq(·) (u(t, ·)) − Q(t). hut (t, ·), u(t, ·)i ≥ kut (t, ·)k22 + γp+ H (t) + dt q− By (4.3.11), Proposition 4.5.1 and Lemma 4.5.3–(iii), we have c˜1 > min{(γp+ w0 /c∞ )1/q− , (γp+ w0 /c∞ )1/q+ },

115

so that ku(t, ·)kq(·) ≥ C1

for all t ∈ I,

(4.6.6)

with C1 given in (4.6.3). Furthermore, (4.3.10) holds, with q1 = (1 + |Ω|)1+κ/m by (4.4.2), applying Young’s inequality, with ` = 2/L. Therefore, using (4.3.10) and (4.6.6), we get (4.3.12), that is n 0 1+κ/m α1 −α2 Lm /m δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) Du(t) Q(t) ≤(1 + |Ω|) C1 o q− 2 , ku(t, ·)kαq(·) +`ku(t, ·)kq(·) being here q2 = (1 + |Ω|)1+κ/m max{1, C1α1 −α2 } = (1 + |Ω|)1+κ/m C1α1 −α2 , 0 0 0 since C1 ≤ 1 and α1 − α2 ≤ 0, and `˜ = max{Lm /m , L℘ /℘ } = Lm /m , since 1 < m ≤ ℘ and L ≥ 1. Being υ(0) > υ0 and Eu(0) < E0 , Lemma 4.5.3 implies that Eu(0) < E0 ≤ E1 < E2 , so that certainly 0 < H0 = E0 − Eu(0) < E2 − Eu(0). On the other hand, condition (4.3.15) continues to hold with c1 = c∞ /q− , cf. Proposition 4.5.1, so that we obtain (4.3.16)–(4.3.18) in I, that is n 0 Q(t) ≤ C2 Lm /m δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) Du(t) o q− [H (t)]−rq− /q+ +`ku(t, ·)kq(·) n 0 r−rq /q ≤ C2 Lm /m H0 − + δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) · o −rq /q q− · [H (t)]−r Du(t) + `H0 − + ku(t, ·)kq(·) where C2 = (1 + |Ω|)1+κ/m C1α1 −α2 (c∞ /γp+ )rq− /q+ max{1, (c∞ /γp+ w0 )r } as in (4.6.3). Let λ > 0 be as in (4.6.4) and define 1,1 Z (t) = λk(t) [H (t)]1−r + hu(t, ·), ut (t, ·)i ∈ Wloc (I),

where r = θq− /(1 + θ)q+ . Of course Z0 = Z (0) > 0 by the choice of λ in (4.6.4). Clearly, for a.a. t ∈ I, it results Z 0 = λk(1 − r)H −r H 0 + λk 0 H 1−r +

d hut , ui, dt

116


and consequently Z 0 ≥ λk(1 − r)H −r H 0 + λk 0 H 1−r cε0 + kut k22 + γp+ H (t) + ρq(·) (u) − Q(t) q− n h io 0 r−rq /q 1/(m−1) 1/(℘−1) ≥ λk(1 − r) − C2 Lm /mH0 − + δ1 + δ2 H −r H 0 + γp+ H (t)+λk 0 H 1−r + kut k22 +

cε0 −rq /q q− ρq(·) (u) − C2 `H0 − + kukq(·) q−

Thus, by (4.6.1), (4.6.6) and the fact that λk 0 H 1−r ≥ 0, we find a.e. in I, h i r−rq− /q+ 0 m0 /m Z ≥k λ(1 − r) − C2 L H0 H −r H 0 + γp+ H (t) ! q+ −q− cε C 0 −rq /q q− 1 − + . + kut k22 + kukq(·) − C2 `H0 q− Note that the quantity in the square brackets is non-negative, because by (4.6.4), it results 0

λ≥

C2 Lm /m r−rq− /q+ H0 1−r

being r ≤ r. Therefore o n q− 2 0 Z (t) ≥ C3 H (t) + kut (t, ·)k2 + ku(t, ·)kq(·) > 0,

(4.6.7)

q −q

where C3 = min{cε0 C1 + − /2q− , 1} > 0, since 2/` = L and by assumption (4.6.4). On the other hand, in place of (4.3.21), we now have α Z α ≤ λkH 1/α + |hut , ui| ≤ 2α−1 {(λk)α H + kut kα2 kukα2 } , where α = q− /(q− − rq+ ) and α ∈ (1, 2) by (4.3.17) and the choice of r. Put ν = 2/α and ν 0 the conjugate exponent of ν. Therefore, as in (4.3.22), for all t ∈ I 0

q

− ku(t, ·)kαν ≤ C4 ku(t, ·)kq(·) , 2

(4.6.8)

117 0

q −αν 0

, being c˜1 > C1 and αν 0 ≤ q− . Hence, in with C4 = (1 + |Ω|)αν /C1 − place of (4.3.23), by Young’s inequality, we have for a.a. t ∈ I o n q− , Z (t)α ≤ C5 [max{λk(t), 1}]α H (t) + kut (t, ·)k22 + ku(t, ·)kq(·) where C5 = 2α−1 C4 . Now recall that k(t) ≥ k0 > 0 for all t ∈ I, so that by the choice of λ in (4.6.4) it follows that α o k(t) n q− α 2 Z (t) ≤ C5 max λk(t), H (t)+kut (t, ·)k2 +ku(t, ·)kq(·) k0 (4.6.9) o n q− α 2 ≤ C5 [λk(t)] H (t) + kut (t, ·)k2 + ku(t, ·)kq(·) . Combining (4.6.9) with (4.6.7), we obtain a.e. in I Z −α Z 0 ≥

C3 (λk)−α . C5

Hence, recalling that α = 1 + θ and noticing that K = C3 /C5 , it follows that for a.a. t ∈ I Z (t)θ ≥

1 Z0−θ

− θKλ−(1+θ)

Rt 0

k(τ )−(1+θ) dτ

= Φ(t).

(4.6.10)

Therefore, Φ(t) % ∞ as t % T0 , where T0 is defined in (4.6.5). Hence Z cannot be continued after T0 , that is u cannot be global and T ≤ T0 , as required. 2 1−θq /[(1+θ)q ]

− + Remark 4.6.2. (i) The request λ ≥ 2hu0 , u1 i− /k0 H0 in (4.6.4) guarantees that Z0 > 0, in the more subtle case hu0 , u1 i < 0. On the other hand, if u0 and u1 are cooperative, that is hu0 , u1 i ≥ 0, then Z0 > 0, being λ > 0 by (4.6.4). Hence, for cooperative data, condition (4.6.4) on λ reduces simply to ( ) (1 + θ)q+ 1 0 r−rq /q λ = max C2 Lm /m H0 − + , , (1 + θ)q+ − θq− k0

where r and r are defined in (4.6.4) and 0 < r ≤ rq− /q+ , see (4.3.17) and comments there.

118


(ii) In the limit case Eu(0) = E0 the expression of H0 > 0 given in (4.6.4), which is independent of the solution, simply becomes zero. (iii) The delicate argument of the proof of Theorem 4.6.1 provides global non-existence of solutions of (4.1.1), but it does not by itself establish that maximal solutions blow up at the lifespan T . On the other hand, global non-existence occurs by the blow up of natural norms, when either T = T0 or limt→T − Z (t) = ∞, as it will be shown in the corollary below. For a pioneering discussion on this phenomenon we refer to [16], as well as to the Introduction. Corollary 4.6.3. Under the assumptions and notations of Theorem 4.6.1, if either T = T0 or limt→T − Z (t) = ∞, then lim ku(t, ·)kq(·) = ∞

t→T −

and

lim ku(t, ·)k = ∞.

t→T −

(4.6.11)

Proof. The proof of Theorem 4.6.1 can be repeated word by word. Hence, by (4.6.10) we get limt→T − Z (t) = ∞ in both cases. Now, relations (4.3.7), (B)–(ii) and (E ) imply that for all t ∈ I H0 ≤ H (t) ≤ H0 + Eu(0) − Eu(t) ≤ H0 + Eu(0) − 12 kut (t, ·)k22 + F u(t). Hence, kut (t, ·)k22 ≤ 2[Eu(0) + F u(t)] < 2[E0 + F u(t)]. Moreover, condition (4.3.13) still holds, so that, by (4.5.5), we get c∞ q− F u(t) ≤ ρq(·) (u(t, ·)), (4.6.12) H (t) ≤ γp+ γp+ and (4.6.9) becomes n o q− Z (t)α ≤ C5 [λk(t)]α H (t) + 2F u(t) + 2E0 + ku(t, ·)kq(·) 2 1 α + ≤ C5 [λk(T )] c∞ ρq(·) (u(t, ·)) γp+ q− 2E0 q− + + 1 ku(t, ·)kq(·) q c˜1− 2 2E0 1 α + + q− + 1 · ≤ C5 [λk(T )] c∞ γp+ q− c˜ n o 1 q+ q− · max ku(t, ·)kq(·) , ku(t, ·)kq(·) ,

119

by (4.6.6), (4.3.11), (1.1.4) and the monotonicity of k > 0, being T ≤ T0 < ∞. Therefore, o n q− q+ ≥ ΛZ (t)α , (4.6.13) , ku(t, ·)kq(·) max ku(t, ·)kq(·) q

1)}. Hence, where Λ = 1/ {C5 [λk(T )]α (c∞ /γp+n+ 2c∞ /q− + 2E0 /˜ c1− + o q− q+ there exists t < T so large that max ku(t, ·)kq(·) , ku(t, ·)kq(·) ≥ 1 for all t ∈ [t, T ), being limt→T − Z (t) = ∞. Thus, n o q+ q− q+ max ku(t, ·)kq(·) , ku(t, ·)kq(·) = ku(t, ·)kq(·)

for all t ∈ [t, T )

and so, by (4.6.13), limt→T − ku(t, ·)kq(·) = ∞. Finally, by the continuity L,p(·) of the embedding W0 (Ω) ,→ Lq(·) (Ω), we have lim ku(t, ·)k = lim− kDL u(t, ·)kp(·) ≥ lim− Sq0 + ku(t, ·)kq(·) = ∞,

t→T −

t→T

t→T

where Sq0 + is given in the statement of Proposition 4.5.2.

2

Of course limt→T − Z (t) ≤ ∞ by (4.6.7) and (4.6.10). If that limit is infinite, a case which occurs when T = T0 , then (4.6.11) is valid as shown in Corollary 4.6.3. While, if limt→T − Z (t) = ZT < ∞, so that T < T0 by Corollary 4.6.3, it could happen that lim supt→T − kDL u(t, ·)kp(·) < ∞, as explained in the Remark 4.6.2–(iii). In this case, or even when lim inf t→T − kDL u(t, ·)kp(·) is finite, we have limt→T − H (t) < ∞. Otherwise, limt→T − H (t) = ∞ by the definition of H and so, by (4.6.12), limt→T − ku(t, ·)kq(·) = ∞. This is clearly impossible by the Sobolev embedding, being q subcritical by (4.5.2). Therefore, the main dynamical part Du of the damped system, the so called damping rate, is actually in L1 (I), I = [0, T ), and this means that the total damping over the entire time interval I is finite. We conclude the section giving some simpler expressions for T0 , when Q satisfies (4.4.1) for all t ∈ R+ 0 and the special case (4.4.3) of (4.6.1), when 0 ≤ s ≤ m − 1, so that ψ ≡ 1 in R+ 0 , see the proof of Proposition 4.4.1. Hence (4.6.2) holds by (4.4.4). Corollary 4.6.4. Let Q be as above and let θ0 be the number given in (4.3.3). If u ∈ X is a solution of (4.1.1) satisfying (4.5.21), then T ≤ T0

120


and

T0 =

 θ λK λK    = Θ0 , if s = 0 and 0 < θ ≤ θ0 ,    θK Z0 eΘ0 − 1,       (mΘ0 + 1)1/m − 1,

if s > 0 and θ = (m − 1 − s)/s, if s > 0 and θ < (m − 1 − s)/s,

where m = [m − 1 − s(1 + θ)]/(m − 1) > 0 being θ < (m − 1 − s)/s, and the constants λ and Z0 are given in (4.6.4). Proof. If s = 0, it is enough to apply Theorem 4.6.1, so that from (4.6.5) we get the claim. Consider now the case 0 < s < m − 1. When 0 < θ ≤ min{θ0 , (m − 1 − s)/s}, condition (4.6.2) holds and again it is possible to apply Theorem 4.6.1, obtaining the above expression of T0 by (4.6.5). 2 Clearly, Corollary 4.6.4 covers all the cases when Q(t, x, u, ut ) = |ut |m−2 ut , as it usually happens in the literature. In particular, Corollary 4.6.4 extends the recent Theorems 4.3 and 4.5 of [107], in which the authors consider a simplified version of (4.1.1) with, in our notations, L = 1, µ = 0, g ≡ 0 and c ≡ 1 in (4.5.2), and the nonlinear damping function Q as in Corollary 4.6.4, with d1 ≡ 0 and d2 ≡ Const.

Chapter 5

Two Kirchhoff-Love Models In this chapter we analyze some strongly damped Kirchhoff systems involving the polyharmonic operator.

5.1

Introduction

In mechanical engineering we often encounter structures composed of rigid and elastic components. The elastic properties of a material depend on the bonding type, crystallinity and composition of its particles. In dimension two the study of plates, membranes and beams is strictly related to the damping technology. This investigation is devoted to the problem of suppression or reduction of vibrations in elastic bodies subjected to forces and moments, as it occurs in the Kirchhoff-Love theory, which is an extension of the Euler-Bernoulli beam theory. The Kirchhoff-Love theory was developed by Love in 1888 under the assumptions proposed by Kirchhoff. It was born for infinitesimal strains and then adapted by von Kármán to manage also moderate rotations, cf. [78]. Mathematically, the phenomena studied in the above theories are modeled by nonlinear differential equations, associated with dissipative and driving forces and governed by the biharmonic operator. Recently, Monneau considered in [85, 86] a three dimensional plate in the framework of isotropic homogeneous nonlinear elasticity for a St. Venant-Kirchhoff material, proving the existence of a uniformly εrescaled solution which converges to the solution of the Kirchhoff-Love plate model as ε → 0; see also the specialistic monograph [30] for a

122

Chapter 5. Two Kirchhoff-Love Models

derivation by asymptotic analysis and the original work by Ciarlet and Destuynder [31], involving semilinear elliptic partial differential equations. In the Kirchhoff-Love plate models, the external damping Q depends only on (t, v) and is linear in v, so that for brevity it is simply denoted by Q(t)v. For L = 1, 2, . . . the problem we consider here is  2 L 2 L−1 u +M kD u(t, ·)k (−∆) u+N kD u(t, ·)k u  L L−1 tt 2 2 (−∆)   L + µu + %(t)(−∆) ut +Q(t)ut = f (t, x, u), (5.1.1)  α  D u(t, x) + =0 for all α, with |α| ≤ L − 1. R0 ×∂Ω

In the whole chapter, Ω ⊂ Rn is a bounded domain, n ≥ 1, the function u = (u1 , . . . , ud ) = u(t, x) is the vectorial displacement, d ≥ 1, R+ 0 = [0, ∞), α is a multi-index, µ is a non-negative parameter and DL is defined in (1.2.2) and has to be intended component by component, as usual. Similarly, by (−∆)L u we mean the d-vector ((−∆)L u1 , . . . , (−∆)L ud ). For (5.1.1), the Kirchhoff functions M and N verify the same assumptions required for problem (4.1.1), namely, M 6≡ 0 and satisfies (M ) M ∈ L1loc (R+ 0 ) is non-negative and there exists γ ≥ 1 such that Z τ + γM (τ ) ≥ τ M (τ ), τ ∈ R0 , M (z)dz, where M (τ ) = 0

N is possibly trivial and verifies condition (M ), with parameter η ∈ [1, γ], Rτ + that is ηN (τ ) ≥ τ N (τ ) for all τ ∈ R0 , where N (τ ) = 0 N (z)dz. Furthermore, we assume Q ∈ C 1 (R+ 0 ),

Q, −Q0 ≥ 0 in R+ 0,

(5.1.2)

which is clearly verified when Q does not depend on t, as it usually happens in literature, see e.g. [2, 57, 59, 90, 107, 108, 109]. Moreover, being Q(t, x, u, v) = Q(t)v, when (5.1.2) holds in the nontrivial case Q(0) > 0, then condition (Q) given in Section 4.3 is trivially verified, with m = ℘ = 2, κ = 0, δ1 (t) = Q(t)|Ω|1/℘1 , ℘1 = q/(q − 2), δ2 ≡ 0, ψ ≡ 1 and k ≡ K, where K ≥ Q(0)|Ω|1/℘1 . In particular, (4.3.1)–(4.3.3) hold for any θ ∈ (0, θ0 ], and the optimal value is θ = θ0 . With this choice of δ1 and δ2 , it is clear that the hypothesis (5.1.2) implies the validity of condition (4.4.3) in the whole R+ 0 , with s = 0, being Q non-increasing.

123

It is well known that %(t)(−∆)L ut represents the internal material damping of Kelvin-Voigt type of the body structure, which is always present, even if small, in real material as long as the system vibrates, see e.g. [54]. Also on % we suppose the natural restriction % ∈ C 1 (R+ 0 ),

%, −%0 ≥ 0 in R+ 0,

(5.1.3)

which is automatic in the standard case in which %(t) ≡ %0 > 0 in R+ 0. The main general result for (5.1.1) is Theorem 5.3.1, in which an explicit a priori estimate T0 for the lifespan time T is established when the initial energy is bounded above by either E1 if γ = 1, or E2 if γ > 1, where γ ≥ 1 is the parameter corresponding to M in (M ) and E1 , E2 are two value indicated in Figure 4.1. Clearly, Theorem 5.3.1 is a noncontinuation result and implies that problem (5.1.1) does not admit any solutions which are global in time. We note that, when % ≡ 0, system (5.1.1) is a special type of (4.1.1) with p ≡ 2 and Theorem 5.3.1 extends Theorem 4.3.1, since the nonlinearity f is assumed more general. This relies on a simpler proof technique which can be used when the damping is linear in the main dynamic variable v. Moreover, Theorem 5.3.1 extends Theorem 5.3 of [109], given for L = 2, M ≡ Const. > 0, µ = 0, Q ≡ 0 and f = f (u). When the initial data belong to the region Σ0 shown in Figure 4.1 and Eu(0) = E0 , concrete applications of Theorem 5.3.1 are given. In particular, Theorem 5.3.2 generalizes the recent Theorem 2.4 of [29], stated for negative initial energy and cooperative initial data. In the same setting, the second model we study for L = 1, 2, . . . is the following strongly damped polyharmonic Kirchhoff system in R+ 0 ×Ω  utt +M kDL u(t, ·)k22 (−∆)L u + N kDL−1 u(t, ·)k22 (−∆)L−1 u    + µu + %(t)K kDL u(t, ·)k22 (−∆)L ut +Q(t, x, u, ut ) = f (t, x, u),    Dα u(t, x) + =0 for all α, with |α| ≤ L − 1. R0 ×∂Ω

(5.1.4) The main Kirchhoff function M has the form M (τ ) = a + bγτ γ−1 ,

a, b ≥ 0,

a + b > 0,

γ > 1 if b > 0, (5.1.5)

where we takeRγ = 1 if b = 0, so that γM (τ ) ≥ τ M (τ ) for all τ ∈ R+ 0, τ with M (τ ) = 0 M (z)dz; while N satisfies

124


(N ) N ∈ L1loc (R+ 0 ) is non-negative and there exists η ∈ [1, γ] such that Z τ + N (z)dz. ηN (τ ) ≥ τ N (τ ), τ ∈ R0 , where N (τ ) = 0

The Kirchhoff function K ∈ L1loc (R+ 0 ) is such that K(0) ≥ 0 and

K(τ ) > 0 for all τ > 0,

(5.1.6)

and is related to M by the assumption (K ) for every σ > 0 there is cσ > 0 such that for all τ ≥ σ Z τ K(z)dz. M (τ ) ≥ cσ K (τ ), where K (τ ) = 0

In particular, relation (K ) is verified when K ≡ 1, while we do not include for this problem the simpler case K ≡ 0, which has been already treated in Chapter 4 of this thesis as well as in Section 3 of [9]. When K ≡ 0 we have given some explicit a priori estimates for the lifespan of local solutions, hence we have proved that no global solutions exist. Conversely, in this section we could expect the solutions to be global in time and we analyze their behaviour at infinity. Similar results for related problems under dynamical boundary conditions can be found e.g. in [11, 12, 55]. The nonlinear external damping Q represents the most common suppressions of the vibrations of an elastic structure of passive viscous type d d and absorbs vibration energy. The function Q ∈ C(R+ 0 ×Ω×R ×R → Rd ) satisfies (4.1.2), that is (Q(t, x, u, v), v) ≥ 0 for all t, x, u, v. The term %(t)K(kDL uk22 )(−∆)L ut represents the internal material damping of Kelvin-Voigt type, which strictly depends on the body structure. Usually, in standard literature, % is a positive constant, see [57, 87, 108, 110], while for (5.1.4) we assume only the mild request % ∈ C 1 (R+ 0 ),

% 6≡ 0 and %, %0 ≥ 0 in R+ 0.

(5.1.7)

In the case % ≡ 0 we know, by Sections 3 and 6 of [9] and by Chapter 4 of this thesis, that problem (5.1.4) has no global solutions. In this

125

chapter we give some results of blow up at infinity of global solutions of (5.1.4), therefore we consider here only the case in which %(t) > 0 for t sufficiently large. d d For both the models, the function f ∈ C(R+ 0 × Ω × R → R ) is an external source force derivable from a potential F , that is (4.1.3) holds, namely f (t, x, u) = Du F (t, x, u), F (t, x, 0) = 0. The first result for (5.1.4) is Theorem 5.4.2, in which the blow up at infinity of global solutions of the system is proved, when Eu(0) is bounded from above by a critical value E1 , see Figure 4.1. In Theorem 5.4.2 and its consequences, the L∞ -norm of % could also be infinity, and anyhow the results here obtained are new even when k%k∞ < ∞. Several applications of Theorem 5.4.2 are given, especially in Theorem 5.4.5, when ku(0, ·)kq > υ0 and Eu(0) ≤ E0 , see (5.2.9)–(5.2.10) for the exact meaning of υ0 and E0 , where k · kq is the Lq -norm and q is a parameter related to the growth of f in u. In particular, in the limit case Eu(0) = E0 , we cover also the case Q ≡ 0 not allowed e.g. in [108] when L = 1. This is possible either when f significantly depends on t or assuming %(0) > 0 and K satisfying an additional growth condition, so that, in the latter case, the presence of an internal dissipation balances the absence of an external damping. We note that model (5.1.4) reduces to a special case of (4.1.1) with p ≡ 2, when %(t)K(kDL u(t, ·)k22 ) ≡ 0 in R+ 0 . In this case, under suitable assumptions on f and Q, Theorem 4.3.1 shows that no global solutions u of (5.1.4) exist, provided that the initial energy Eu(0) associated to the system is appropriately bounded above by the critical value E2 . This is possible thanks to a certain interaction between the external force and the damping term. Therefore, a fairly natural question was to understand what happens if the additional damping %(t)K(kDL u(t, ·)k22 ) (−∆)L ut is included in the system. In [40] D’Ancona and Shibata study the global existence of analytic solutions of problems describing nonlinear viscoelastic materials with short memory, which are a special case of (5.1.4) when L = 1, N ≡ 0, % ≡ 1, Q ≡ 0 and f ≡ 0. We refer to [15] for a wide list of references. On the other hand, in [87], Ono proved the existence of a global solution u of a subcase of (5.1.4), with Q ≡ 0, % ≡ 1 and K ≡ 1, assuming Eu(0) limited above and ku(0, ·)kq small enough, obtaining ku(t, ·)kq → 0 as t → ∞, see also [89, 90].

126


Indeed, a strong action of dissipative terms could make easier the existence of global solutions, since they play the role of stabilizing terms and their smoothing effect makes more difficult the blow up. In any case, the function % makes the analysis more delicate even when it helps in obtaining the stability of global solutions. For the stability problem of damped Kirchhoff systems, we refer to [8] for the general case and to [57] for special cases of (5.1.4), when L = 1 and N ≡ 0. We want to stress here that the situation considered in Section 5.3 is a very special case in which (5.1.7) is replaced by the complementary condition (5.1.3), the external damping Q is of the special type Q(t, x, u, v) = d1 (t)v and K ≡ 1. More precisely, in Theorems 5.3.1– 5.3.3, under additional hypotheses on the initial data, we provide some a priori estimates for the lifespan of maximal solutions of (5.1.4), which imply the non-continuation of local solutions. This means that, in the case of linear external damping, Theorem 5.4.2 makes sense only when %0 > 0. For a detailed discussion see the remarks after the proofs of Theorems 5.4.2 and 5.4.5.

5.2

Preliminaries and Energy Estimates

In this section we collect a series of notations and preliminaries used throughout the chapter. Since we are in the vectorial setting, we consider maps assuming values in Rd , endowed with the Euclidean norm | · |d . Moreover, since we are in the Hilbert case p ≡ 2, we endow the Sobolev space involved in the treatment, with the norm arising from the scalar product. Hence, the vectorial space [L2 (Ω)]d is equipped with the norm !1/2 !1/2 Z X d d X kψk[L2 (Ω)]d = k |ψ|d k2 = ψi2 = kψi k22 Ω i=1

i=1 d

for every vector valued map ψ : Ω → R , with ψ = (ψ1 , . . . , ψd ), where k · k2 denotes the standard norm of the Lebesgue scalar space L2 (Ω). Similarly, for every L = 1, 2, . . . the vectorial space [H L (Ω)]d is the classical Sobolev space [W L,2 (Ω)]d , endowed with the norm  1/2 X kψk[H L (Ω)]d =  k |Dα ψ|d k22  , |α|≤L

127

where Dα denotes the vectorial |α|-order derivative applied component by component, defined in Section 1.3. The space [H0L (Ω)]d is the completion of [C0∞ (Ω)]d , with respect to the norm k · k[H L (Ω)]d . An equivalent norm for [H0L (Ω)]d is given by  1/2 X kψk[H0L (Ω)]d =  k |Dα ψ|d k22  . |α|=L

Indeed, since Ω is a bounded domain, there exists a positive constant CL = CL (Ω) such that for all ψ ∈ [H0L (Ω)]d kψk2[H L (Ω)]d

=

d X

kψi k2H L (Ω)

i=1

≤ CL

d X X

kDα ψi k22 = CL kψk2[H L (Ω)]d ,

i=1 |α|=L

by Poincaré’s inequality, see for example [1, 6.26], with m = L, p = 2. By integrating by parts, we get kψk[H0L (Ω)]d = k |DL ψ|s k2 for all ψ ∈ [C0∞ (Ω)]d , where DL ψ = (DL ψ1 , . . . , DL ψd ), DL ψi is defined in (1.2.2) and s = d if L is even, while s = nd if L is odd. Finally, by a density argument, it is easy to see that the equality holds also for all ψ ∈ [H0L (Ω)]d . In what follows we endow the space [H0L (Ω)]d with the inner product Z for all ϕ, ψ ∈ [H0L (Ω)]d , (ϕ, ψ)L = (DL ϕ, DL ψ)s dx Ω

where the symbol (·, ·)s denotes the Euclidean inner product of Rd if L is even or Rnd if L is odd. This inner product generates the norm k·k = k |DL ·|s k2 adopted throughout the chapter for the space [H0L (Ω)]d . From now on, we delete the subscript s in (·, ·)s . The elementary R bracket pairing hϕ, ψi = Ω (ϕ(x), ψ(x))dx is well defined for all ϕ, ψ such that (ϕ, ψ) ∈ L1 (Ω). For simplicity in notation we drop the exponents d and nd in all the functional spaces involved in the treatment, thus H0L (Ω) denotes [H0L (Ω)]d , and L2 (Ω) = (L2 (Ω), k·k2 ) is used in all the dimensions 1, d and nd. In this final part of the section we treat both the models simultaneously and we unify the notations denoting the lifespan of the maximal solutions by ( ( T ∈ (0, ∞] for (5.1.1), [0, T ) for (5.1.1), T = hence I = [0, T ) = ∞ for (5.1.4), R+ for (5.1.4). 0

128


For both the models (5.1.1) and (5.1.4), the main solution and test function space is X = C 1 (I → H0L (Ω)) For every φ ∈ X the functional A , which includes the elliptic part of the systems, coincides with the A defined in Section 4.2 for (4.1.1), when p ≡ 2 and is given by (A ) A φ(t) = 12 M (kDL φ(t, ·)k22 ) + N (kDL−1 φ(t, ·)k22 ) + µkφ(t, ·)k22 , so that A φ is the Fréchet potential, with respect to φ, of the operator Aφ, defined pointwise for all (t, x) ∈ I × Ω by Aφ(t, x) = M (kDL φ(t, ·)k22 )(−∆)L φ(t, x) + N (kDL−1 φ(t, ·)k22 )(−∆)L−1 φ(t, x) + µφ(t, x). Clearly A φ ≥ 0 in I × Ω, being M and N non-negative and µ ≥ 0. Moreover for all u, φ ∈ X and t ∈ I we have hAu(t, ·), φ(t, ·)i =M (kDL u(t, ·)k22 )hDL u(t, ·), DL φ(t, ·)i+µhu(t, ·), φ(t, ·)i + N (kDL−1 u(t, ·)k22 )hDL−1 u(t, ·), DL−1 φ(t, ·)i. In particular, hAφ(t, ·), φ(t, ·)i = M (kDL φ(t, ·)k22 )kDL φ(t, ·)k22 + µkφ(t, ·)k22 + N (kDL−1 φ(t, ·)k22 )kDL−1 φ(t, ·)k22 , so that hAφ(t, ·), φ(t, ·)i ≤ 2γA φ(t) for all (t, φ) ∈ I × X,

(5.2.1)

by (5.1.5) and (N ). On the external force f we require the usual structural assumptions (F1 )

(F2 )

F (t, ·, φ(t, ·)), (f (t, ·, φ(t, ·)), φ(t, ·)) ∈ L1 (Ω) for all t ∈ I, φ ∈ X, hf (t, ·, φ(t, ·)), φ(t, ·)i ∈ L1loc (I) for all φ ∈ X, L Ft ≥ 0 in R+ 0 × H0 (Ω),

129

R where F φ(t) = F (t, φ) = Ω F (t, x, φ(t, x)) dx is the potential energy of the field φ ∈ X, which is well defined by (F1 ), and Ft denotes the partial L derivative with respect to t of F = F (t, w) for (t, w) ∈ R+ 0 × H0 (Ω). The natural total energy of the field φ ∈ X, associated to both (5.1.1) and (5.1.4) is Eφ(t) = 12 kφt (t, ·)k22 + A φ(t) − F φ(t),

(E )

and it is well defined in X by (F1 ). We say that a solution of either (5.1.1) or (5.1.4) is a function u ∈ X satisfying the following properties (A) and (B): (A) Distribution Identity for (5.1.1) Z tn t hut , φi 0 = hut , φt i − hAu, φi − %(τ )hDL ut , DL φi 0 o − hQ(τ )ut − f (τ, ·, u), φ(τ, ·)i dτ for all t ∈ I and φ ∈ X; (A) Distribution Identity for (5.1.4) Z tn t hut , φt i − hAu, φi − %(τ )K(kDL uk22 )hDL ut , DL φi hut , φi 0 = 0 o − hQ(τ, ·, u, ut ) − f (τ, ·, u), φ(τ, ·)i dτ for all t ∈ I and φ ∈ X; (B) Energy Conservation 1 Du(t) Z t∈ Lloc (I), (ii) Eu(t) ≤ Eu(0) − Du(τ )dτ for all t ∈ I,

(i)

0

where Du(t) = hQ(t, ·, u, ut ), ut i + %(t)K(kDL u(t, ·)k22 )kDL ut (t, ·)k22 + Ft u(t), ( 1 for (5.1.1), K(τ ) = for all τ ∈ R+ 0. K(τ ) for (5.1.4),

130


The Distribution Identity is meaningful provided that hf (t, ·, u), φi ∈ L1loc (I) and hQ(t, ·, u, ut ), φi ∈ L1loc (I), along the field φ ∈ X. The first condition is valid whenever (F1 ) is in charge, while the latter is assumed. These restrictions are satisfied in the special cases treated in the theorems, as well as in the applications. The other terms in the Distribution Identity are well defined thanks to the choice of the space X. Let u ∈ X be a solution of either (5.1.1) or (5.1.4), we put for convenience w1 = inf A u(t), w2 = inf F u(t), t∈I t∈I (5.2.2) 2γ q w1 , E2 = − 1 w2 . E1 = 1 − q 2γ Let us introduce these other two conditions on f . (F3 )0 There exists a number q such that q > 2γ,

(5.2.3)

and for all F > 0 and φ ∈ X for which inf t∈I F φ(t) ≥ F, it results hf (t, ·, φ(t, ·)), φ(t, ·)i ≥ qF φ(t) for all t ∈ I. (F4 )0 There exists a number q verifying (5.2.3) and for all F > 0 and φ ∈ X for which inf t∈I F φ(t) ≥ F, there exist c1 = c1 (F, φ) > 0 and ε0 = ε0 (F, φ) > 0 such that (i)

F φ(t) ≤ c1 kφ(t, ·)kqq for all t ∈ I,

and for all ε ∈ (0, ε0 ] there exists c2 = c2 (F, φ, ε) > 0 such that (ii) hf (t, ·, φ(t, ·)), φ(t, ·)i − (q − ε)F φ(t) ≥ c2 kφ(t, ·)kqq for all t ∈ I. We point out that condition (F3 )0 coincides with the assumption (F3 ) given in Section 4.2, when p ≡ 2 and q ≡ Const. Moreover, (F4 )0 –(ii) implies hf (t, ·, φ(t, ·)), φ(t, ·)i − (q − ε)F φ(t) ≥ 0, so that letting ε → 0 we get (F3 )0 . From now on in this section we require, without further mentioning ( > 1, if b > 0, M (τ ) = a+bγτ γ−1 , a, b ≥ 0, a+b > 0, γ (5.2.4) = 1, if b = 0,

131

f (t, x, u) = g(t, x)|u|σ−2 u + c(x)|u|q−2 u, 1 ≤ σ ≤ q, ( ≤ 2∗L = 2n/(n − 2L), if n > 2L, 2γ < q < ∞, if 1 ≤ n ≤ 2L,

(5.2.5)

where c ∈ L∞ (Ω) is a non-negative function such that c∞ = kck∞ > 0; + while g ∈ C(R+ 0 ×Ω), differentiable with respect to t, with gt ∈ C(R0 ×Ω), satisfies 1 0 ≤ −g(t, x), gt (t, x) ≤ h(x) in R+ 0 × Ω, for some h ∈ L (Ω), ( (5.2.6) q/(q − σ), if σ < q, g(t, ·) ∈ Lη (Ω) in R+ 0 , where η = ∞, if σ = q.

Note that conditions (5.2.5) and (5.2.6) of Section 4.5 reduce to (4.5.2) and (4.5.3), respectively, when p ≡ 2, σ ≡ Const. and q ≡ Const. The only difference is that in (5.2.5), σ and q are allowed to be 1, while in the setting of the variable exponent Lebesgue spaces, q− , σ− > 1. The following two results are the counterpart of Propositions 4.5.1 and 4.5.2, when p ≡ 2 and q ≡ Const. We state them for the sake of clarity. Proposition 5.2.1. If f is of the type given in (5.2.5), then conditions (F1 )–(F2 ), (F3 )0 and (F4 )0 –(i) are satisfied with c1 = c∞ /q. From now on in the section, we fix a solution u ∈ X of either (5.1.1) or (5.1.4) and in correspondence to u we define υ(t) = ku(t, ·)kq . It is clear, from the definitions given above, that (5.1.1) and (5.1.4) have the same elliptic part A and source force f . The energy estimate we present below is related only to A and F , and therefore it is valid for both the systems. Proposition 5.2.2. The following inequality holds for all t ∈ I Eu(t) ≥ ϕ(υ(t)) =

s(Sq0 )2γ c∞ 2γ υ(t) − υ(t)q , γ 2 q

(5.2.7)

where Sq0 = 1/κL Sq , Sq is the Sobolev constant of the embedding H0L (Ω) ,→ Lq (Ω), κL is given in Proposition 1.2.2, s = b if b > 0 and s = a if b = 0.

132


Proof. Following the proof of Proposition 4.5.2, we have A u(t) ≥

s(Sq0 )2γ υ(t)2γ 2γ

and F u(t) ≤

c∞ υ(t)q q

(5.2.8) 2

for all t ∈ I and we conclude. Also here, it is easy to check that ϕ attains its maximum at 1/(q−2γ)

υ 0 = a0

,

a0 =

sγ(Sq0 )2γ , c∞ 2γ−1

(5.2.9)

which coincides with υ0 defined in (4.5.10) for p ≡ 2 and q ≡ Const., and we give the following definitions s(Sq0 )2γ υ02γ 2γ w0 = E0 > 0, w0 = > 0, ϕ(υ0 ) = 1 − (5.2.10) q 2γ 2 Σ0 = {(υ, E) ∈ R : υ > υ0 , E < E0 }. Obviously, also Lemma 4.5.4 is valid in this setting, with p = 2. Moreover consider now the further hypothesis. (D)0 There exists t∗ ∈ (0, T ) such that one of the following properties holds: (i) φ ∈ X and hQ(t, ·, φ, φt ), φt i ≡ 0 in [0, t∗ ] implies either φ(t, ·) ≡ 0 or φt (t, ·) ≡ 0 in [0, t∗ ] and Q verifies for all (t, x, u, v) in R+ 0 ×Ω× Rd × Rd Q(t, x, u, v) = d1 (t, x)|u|κ |v|m−2 v + d2 (t, x, u)|v|℘−2 v,

(5.2.11)

where d1 , d2 and all the exponents are as in (4.4.1); + (ii) there exists a positive function g0 : R+ 0 → R such that gt (t, x) ≥ g0 (t) for each (t, x) ∈ [0, t∗ ] × Ω;

(iii) %(t) > 0 for all t ∈ [0, t∗ ] and K(τ ) ≥ α + βτ δ−1 , τ ≥ 0, α, β ≥ 0, α + β > 0, δ > 1. Condition (5.2.11) in (D)0 –(i) implies that Proposition 4.4.1 and inequality (4.4.2) holds for Q, with q ≡ Const. for all t ∈ R+ 0 . Moreover, in the simpler case Q(t, x, u, v) = Q(t)v, as in (5.1.1), condition (5.2.11) is

133

automatic under the assumption of (5.1.2). The case (D)0 –(iii) is trivially verified when % and K are positive constants and in particular, it reduces only to the request that %(t) > 0 for all t ∈ [0, t∗ ] for problem (5.1.1) in which K ≡ 1. Furthermore, in system (5.1.4), the first part of condition (D)0 –(iii), namely the assumption that %(t) > 0 for all t ∈ [0, t∗ ], is certainly verified when %(0) > 0, being %, %0 ≥ 0. Lemma 5.2.3. Assume (D)0 . Let u ∈ X be a solution of either (5.1.1) or (5.1.4) such that (4.5.17) holds, namely υ(0) > υ0

and

Eu(0) = E0 .

Then υ(I) ⊂ (υ0 , ∞). Furthermore (υ(t), Eu(t)) ∈ Σ0 for all t ∈ (0, T ), that is υ(t) > υ0

and

Eu(t) < E0

for all t ∈ (0, T ).

(5.2.12)

In particular, w1 > w0 , w2 > 2γw1 /q and E0 < E1 < E2 . Proof. Let u be as in the statement. First we show that υ(t) 6= υ0 for all t ∈ I. Proceed by contradiction and suppose that there exists t0 ∈ I such that υ(t0 ) = υ0 . Then, by (5.2.7) and the assumption Eu(0) = E0 , it follows that E0 = Eu(0) ≥ Eu(t0 ) ≥ ϕ(υ(t0 )) = E0 . Hence E0 = Eu(t0 ), so that, by (B)–(ii) and the fact that Du ≥ 0, we get Z t0 E0 = Eu(t0 ) ≤ Eu(0) − Du(t)dt ≤ Eu(0) = E0 . 0

R t0

Therefore 0 Du(t)dt = 0 and in turn Du ≡ 0 in [0, t0 ]. Consequently, by the definition of Du, we have that hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i = 0, Ft u(t) = 0 and %(t)K(kDL u(t, ·)k22 )kDL ut (t, ·)k22 = 0 for all t ∈ [0, s0 ], where s0 = min{t∗ , t0 } and t∗ is the number given in (D)0 . Let us now distinguish three cases. Case (D)0 –(i). Since hQ(t, ·, u(t, ·), ut (t, ·)), ut (t, ·)i = 0 for all t ∈ [0, s0 ], we get that either u(t, ·) = 0 or ut (t, ·) = 0 for all t ∈ [0, s0 ]. The first event cannot occur since υ(0) = ku(0, ·)kq > υ0 > 0 by assumption. In the latter, u is clearly constant with respect to t in

134


[0, s0 ], and so u(t, ·) = u(0, ·) for each t ∈ [0, s0 ]. Taking φ(t, ·) = u(0, ·) in the Distribution Identity (A), for each t ∈ [0, s0 ] we have Rt thAu(0, ·), u(0, ·)i = 0 hf (τ, ·, u(0, ·)), u(0, ·)idτ , since (4.4.2) holds in R+ 0 ×Ω by (5.2.11), Du ≡ 0 in [0, s0 ] and so hQ(t, ·, u(0, ·), 0), u(0, ·)i = 0. Thus hAu(0, ·), u(0, ·)i = hf (t, ·, u(0, ·)), u(0, ·)i for each t ∈ [0, s0 ], and in particular hAu(0, ·), u(0, ·)i = hf (0, ·, u(0, ·)), u(0, ·)i. Now 2γA u(0) ≥ qF u(0) by (5.2.1) and (F3 ). On the other hand, E0 = Eu(0) = A u(0) − F u(0) by (E ), since ut (0, ·) = 0. By (5.2.8) and (4.5.17) we have A u(0) > w0 > 0, and so 2γ 2γ E0 ≥ 1 − A u(0) > 1 − w0 = E0 , q q which is an obvious contradiction. Case (D)0 –(ii). We have Z |u(t, x)|p g0 (t) 0 = Ft u(t) = gt (t, x) dx ≥ ku(t, ·)kpp ≥ 0 p p Ω for each t ∈ [0, s0 ]. Therefore, ku(t, ·)kp ≡ 0 and so u ≡ 0 in [0, s0 ] × Ω. But this occurrence is impossible, since υ(0) = ku(0, ·)kq > υ0 > 0, so that we reach a contradiction. Case (D)0 –(iii). We have 0 = %(t)K(kDL u(t, ·)k22 )kDL ut (t, ·)k22 o n 2(δ−1) kDL ut (t, ·)k22 ≥ 0, ≥ %(t) α + βkDL u(t, ·)k2

(5.2.13)

o n 2(δ−1) kDL ut (t, ·)k22 = 0, being for all t ∈ [0, s0 ], and so α + βkDL u(t, ·)k2 %(t) > 0 for t ∈ [0, s0 ]. We distinguish two cases. If α > 0, then kDL ut (t, ·)k2 = 0 for all t ∈ [0, s0 ]. In this case, clearly kut (t, ·)k2 ≤ S2 kDL ut (t, ·)k2 , being u ∈ X = C 1 (I → H0L (Ω)), where S2 is the Sobolev constant for the embedding H0L (Ω) ,→ L2 (Ω). Thus kut (t, ·)k2 = 0 and consequently ut (t, x) = 0 for all t ∈ [0, s0 ] and x ∈ Ω. Therefore u is constant with respect to t in [0, s0 ], but this case cannot occur as shown above in case (i).

135 2(δ−1)

On the other hand, if α = 0, then kDL u(t, ·)k2 kDL ut (t, ·)k22 = 0 for all t ∈ [0, s0 ], being β > 0. Now d kDL u(t, ·)k22 ≤ 2|hDL ut (t, ·), DL u(t, ·)i| ≤ 2kDL ut (t, ·)k2 kDL u(t, ·)k2 , dt hence kDL u(t, ·)k2 = Const. ≥ 0. If kDL u(t, ·)k2 = Const. > 0, then for all t ∈ [0, s0 ], kDL ut (t, ·)k2 = 0 and we obtain the contradiction following the argument above. Otherwise, kDL u(t, ·)k2 ≡ 0, that is u(t, ·) = 0 for all t ∈ [0, s0 ]. Again this occurrence is impossible, being υ(0) = ku(0, ·)kq > υ0 . Consequently, in all the three cases, υ(t) > υ0 for all t ∈ I by the continuity of t 7→ υ(t), being υ(0) > υ0 by (4.5.17). Now Eu(t) ≤ E0 for all t ∈ I Rand there are no points t0 ∈ (0, T ) t such that Eu(t0 ) = E0 . Otherwise 0 0 Du(t)dt = 0 and assumption (D)0 would provide a contradiction, as shown above. Hence Eu(t) < E0 for all t ∈ (0, T ) and this concludes the proof of (5.2.12). By (5.2.8), we have A u(t) ≥ w1 ≥ w0 for all t ∈ I. The case w1 = w0 cannot occur. Indeed, assume by contradiction that w1 = w0 . First note that A u cannot reach the value w0 at a time t0 before the lifespan T . Indeed, if there is t0 ∈ I such that A u(t0 ) = w0 , then υ(t0 ) ≤ υ0 by (5.2.8) and (5.2.10), which is impossible by (5.2.12). Thus, since inf t∈I A u(t) = w1 = w0 there exists a minimizing sequence (tk )k ⊂ I such that limk→∞ A u(tk ) = w0 and (tk )k cannot be contained in an interval [0, τ0 ], with τ0 < T , since A u reaches its infimum at T . Hence, (tk )k admits a subsequence, still denoted by (tk )k , converging to T as k → ∞. Therefore lim inf t→T − A u(t) ≤ w0 and this forces lim inf t→T − A u(t) = w0 , being w0 = inf t∈I A u(t). Put now υ1 = inf t∈I υ(t), so that A u(t) ≥ sυ(t)2γ /(2Sq2 )γ ≥ sυ12γ /(2Sq2 )γ for all t ∈ I by (5.2.8). Consequently w0 ≥ sυ12γ /(2Sq2 )γ , and in turn υ1 ≤ υ0 by (5.2.9) and (5.2.10). On the other hand, υ1 ≥ υ0 being υ(t) > υ0 for all t ∈ I by (4.5.18) and so υ1 = υ0 . Arguing as for A , we obtain that lim inf t→T − υ(t) = υ0 . Finally, w0 − c∞ υ(t)q /q < A u(t) − F u(t) ≤ Eu(t) ≤ E0 by (E ), (4.5.5) and (B)–(ii), so that lim supt→T − Eu(t) = E0 and consequently, by (B)– Rt RT (ii), (4.1.2) and (F2 ), it follows limt→T − 0 Du(τ )dτ = 0 Du(τ )dτ = 0, so that Du ≡ 0 in I. This is again impossible as shown above. In conclusion w1 > w0 , and so E0 < E1 . Furthermore, F u(t) ≥ w1 − Eu(0) = w1 − E0 > w1 − E1 = 2γw1 /q for all t ∈ I by (E ), so that

136


E1 < E2 . Hence, E0 < E1 < E2 , as required.

2

If in addition f satisfies 1≤σ 0,

(5.2.14)

then also (F4 )0 –(ii) is verified with ε0 ∈ (0, q − σ] and c2 = cε/q for all ε ∈ (0, ε0 ]. To prove the validity of (F4 )0 –(ii), fix φ ∈ X and ε0 ∈ (0, q −σ]. Then by (5.2.5), hf (t, ·, φ(t, ·)), φ(t, ·)i − (q − ε)F u(t) Z Z ε q−ε σ g(t, x)|φ(t, x)| dx + c(x)|φ(t, x)|q dx = 1− σ q Ω Ω cε q ≥ kφ(t, ·)kq , q as claimed.

5.3

Lifespan Estimates for the First Model

We provide now some a priori estimates for the lifespan T of (maximal) solutions of (5.1.1). The importance of such a model relies principally in the applications of mathematical physics. Recently, Wu and Tsai have studied in [109] the existence and non-existence of global solutions of a subcase of (5.1.1). More precisely they consider L = 2, µ = 0, % ≡ 0, M ≡ a > 0, Q(t) ≡ 0 and f only dependent on u. In [109] some blow up properties of solutions have been established by energy methods and lifespan estimates for local solutions obtained by direct methods. The subsequent Theorem 5.3.1 is established by assuming (5.1.2) and (5.1.3). Moreover, we require that f satisfies (F1 )–(F2 ) and (F3 )0 . From now on in the chapter we shall use the following notation u0 = u(0, ·) and u1 = ut (0, ·). Theorem 5.3.1. Let u ∈ X be a solution of (5.1.1) satisfying ( E2 , if γ > 1, Eu(0) < E1 , if γ = 1.

137

Then for all β0 ≥ 0 such that  1   Q(0)ku0 k22 + %(0)kDL u0 k22 , γ > 1, 1 hu0 , u1 i + ββ0 > γ − (5.3.1) 2   Q(0)ku0 k22 + %(0)kDL u0 k22 , γ = 1, q−2 ( 2[E2 − Eu(0)], γ > 1, where β = it results that T ≤ T0 , where 2[E1 − Eu(0)], γ = 1,

T0 =

 ku0 k22 + ββ02   ,    (γ − 1)[hu0 , u1 i + ββ0 ] − [Q(0)ku0 k22 + %(0)kDL u0 k22 ]     

2ku0 k22 + 2ββ02 , (q − 2)[hu0 , u1 i + ββ0 ] − 2 [Q(0)ku0 k22 + %(0)kDL u0 k22 ]

γ > 1,

γ = 1.

In particular, if Q ≡ 0, % ≡ 0 and the initial data are strongly cooperative, that is hu0 , u1 i > 0, then T ≤ T0 , where now  ku0 k22   , γ > 1,    (γ − 1)hu0 , u1 i T0 =   2ku0 k22    , γ = 1. (q − 2)hu0 , u1 i Proof. Let u ∈ X be a solution of (5.1.1) as in the statement. Let us divide the proof into two cases. Case γ > 1. Take β, β0 and T0 be as in the statement and assume by contradiction that T > T0 . The function Z t 2 G (t) = ku(t, ·)k2 + Q(τ )ku(τ, ·)k22 + %(τ )kDL u(τ, ·)k22 dτ 0 Z t 0 + (τ − t) Q (τ )ku(τ, ·)k22 + %0 (τ )kDL u(τ, ·)k22 dτ 0 + (T0 − t) Q(0)ku0 k22 + %(0)kDL u0 k22 + β(t + β0 )2 , is well defined in [0, T0 ] and twice differentiable in [0, T0 ]. Since Q, % ∈

138


C 1 (R+ 0 ), it results G 0 (t) = 2hu(t, ·), ut (t, ·)i + Q(t)ku(t, ·)k22 + %(t)kDL u(t, ·)k22 Z t 0 Q (τ )ku(τ, ·)k22 + %0 (τ )kDL u(τ, ·)k22 dτ − 0

− Q(0)ku0 k22 − %(0)kDL u0 k22 + 2β(t + β0 ) = 2hu(t, ·), ut (t, ·)i + 2β(t + β0 ) Z t [Q(τ )hu(τ, ·), ut (τ, ·)i + %(τ )hDL u(τ, ·), DL ut (τ, ·)i] dτ. +2 0

From the Distribution Identity (A), taking φ = u ∈ X, it follows that 1 2

G 00 (t) = kut (t, ·)k22 − Q(t)hut (t, ·), u(t, ·)i − %(t)hDL ut (t, ·), DL u(t, ·)i − hAu(t, ·), u(t, ·)i + hf (t, ·, u(t, ·)), u(t, ·)i + Q(t)hu(t, ·), ut (t, ·)i + %(t)hDL u(t, ·), DL ut (t, ·)i + β = kut (t, ·)k22 − hAu(t, ·), u(t, ·)i + hf (t, ·, u(t, ·)), u(t, ·)i + β.

Now observe that, thanks to (5.2.1) and (F3 ), we have hAu(t, ·), u(t, ·)i − hf (t, ·, u(t, ·)), u(t, ·)i ≤ 2γA u(t) − qF u(t). Hence, combining these formulas with the definition of the energy function (E ), we get 1 2

G 00 (t) ≥ kut (t, ·)k22 − 2γA u(t) + qF u(t) + β = (1 + γ) kut (t, ·)k22 + (q − 2γ)F u(t) − 2γEu(t) + β.

(5.3.2)

Now (q − 2γ)F u(t) ≥ (q − 2γ)w2 = 2γE2 by (5.2.2), and by the Energy Conservation (B) and (F2 ) also Eu(t) ≤ Eu(0) − I (t), where Z t I (t) = Q(τ )kut (τ, ·)k22 + %(τ )kDL ut (τ, ·)k22 dτ. (5.3.3) 0

Therefore, 1 00 G (t) 2

≥ (1 + γ) kut (t, ·)k22 + β + 2γI (t),

(5.3.4)

being β = 2[E2 − Eu(0)] > 0. Clearly, G 0 (0) = 2hu0 , u1 i + 2ββ0 > 0 and also G (0) = ku0 k22 + T0 [Q(0)ku0 k22 + %(0)kDL u0 k22 ] + ββ02 > 0 by

139

(5.3.1), which assures that hu0 , u1 i > 0 if β0 = 0. Then, since Q and % are non-negative in R+ 0 , it results G 00 , G 0 , G > 0

in [0, T0 ].

(5.3.5)

G G 00 − α G 02 ≥ 0 in [0, T0 ],

(5.3.6)

We assert that where α = (1 + γ)/2. Put A = ku(t, ·)k22 +I (t)+β(t+β0 )2 ,

B = 21 G 0 ,

C = kut (t, ·)k22 +I (t)+β.

Since Q, −Q0 , % and −%0 are non-negative in R+ 0 by (5.1.2) and (5.1.3), we have A ≤ G in [0, T0 ]. (5.3.7) Moreover, by (5.3.4) and the fact that 2γ > 1 + γ, being γ > 1, we get C ≤ G 00 /2(1 + γ) in [0, T0 ].

(5.3.8)

Observe that for all (ξ, η) ∈ R2 and t ∈ [0, T0 ] Aξ 2 + 2Bξη + Cη 2 ηut (t, ·)k22

Z

t

=kξu(t, ·) + + %(τ )kξDL u(τ, ·) + ηDL ut (τ, ·)k22 dτ 0 Z t + Q(τ )kξu(τ, ·) + ηut (τ, ·)k22 dτ + β {(t + β0 )ξ + η}2 ≥ 0, 0 2 because Q and % are non-negative in R+ 0 . Thus AC − B ≥ 0. Hence, (5.3.6) holds by virtue of (5.3.7), (5.3.8) and the fact that A, C > 0. Clearly α > 1 since γ > 1 by assumption. Now (5.3.6) can be written as (G −α G 0 )0 ≥ 0, so that

G 0 (t) G 0 (0) ≥ > 0 for t ∈ [0, T0 ]. G α (t) G α (0) Integrating over [0, t], 0 < t < T0 , and denoting the Hölder conjugate exponent of α by α0 , we have 0

G (0)α G (t) ≥ . [G (0) − (α − 1)G 0 (0)t]1/(α−1)

140


This is impossible, since the left hand side is finite at t = T0 , while the right hand side blows up as t → T0− . Case γ = 1. We proceed exactly as in the previous case with slight changes. In particular, given β, β0 and T0 be as in the statement, assume once more by contradiction that T > T0 and define in [0, T0 ] the same function G given above, obtaining in place of (5.3.2) the estimate G 00 (t) ≥ (2 + q) kut (t, ·)k22 + 2(q − 2)A u(t) − 2qEu(t) + 2β.

(5.3.9)

Using the fact that (q − 2)A u(t) ≥ (q − 2)w1 = qE1 , from the previous relation, in place of (5.3.4), we get G 00 (t) ≥ (2 + q) kut (t, ·)k22 + β + 2qI (t), where now β = 2{E1 − Eu(0)} > 0 and I is defined in (5.3.3). From here on the proof is the same as in the first part, with q in place of 2γ and α = (2 + q)/4. Again α > 1, since q > 2 by (5.2.3), being γ = 1. 2 With Theorem 5.3.1 we extend Theorem 5.3 of [109]. We note in passing that formula (4.10) of [109] is not correct, since, in the notation of [109], the Sobolev constant B1 is not appropriately used. Furthermore we remark that even for linear dampings the case (F3 ) was not covered in [104]–[108], while it first appears in [97], for much simpler problems. The last two theorems are applications of Theorem 5.3.1 and provide an a priori estimate for T , depending only on the initial data. From now on in this section we require, without further mentioning, (5.1.2), (5.1.3) and (5.2.4)–(5.2.6) and we recall the position υ(t) = ku(t, ·)kq , for a fixed solution u ∈ X of (5.1.1). Theorem 5.3.2. Let u ∈ X be a solution of (5.1.1) satisfying ku0 kq > υ0 ,

Eu(0) < E0 .

Put α0 = 2[E0 − Eu(0)] > 0. For all β0 ≥ 0 such that  1  [Q(0)ku0 k22 + %(0)kDL u0 k22 ] , γ > 1,   γ − 1 hu0 , u1 i + α0 β0 >     2 [Q(0)ku0 k22 + %(0)kDL u0 k22 ] , γ = 1, q−2

141

denote by  ku0 k22 + α0 β02   , γ > 1,    (γ − 1)[hu0 , u1 i + α0 β0 ] − [Q(0)ku0 k22 + %(0)kDL u0 k22 ] T0 =   2ku0 k22 + 2α0 β02    , γ = 1. (q − 2)[hu0 , u1 i + α0 β0 ] − 2 [Q(0)ku0 k22 + %(0)kDL u0 k22 ] (5.3.10) Then T ≤ T0 . Proof. Let u ∈ X be as in the statement. The first part of Lemma 4.5.4 and (4.5.21) yield υ(I) ⊂ (υ0 , ∞), being υ(0) > υ0 , in other words (ii) of Lemma 4.5.4 holds. Therefore, Eu(0) < E0 < E1 < E2 by Lemma 4.5.4. We proceed as in the proof of Theorem 5.3.1, but replacing β with α0 in the definition of G . In particular, when γ > 1 from (5.3.2) we now get 1 2

G 00 (t) ≥ (1 + γ) kut (t, ·)k22 + 2γ{E2 − Eu(0)} + α0 + 2γI (t) > (1 + γ) {kut (t, ·)k22 + α0 } + 2γI (t),

where I is defined in (5.3.3). From now on the proof is as before, with T0 given by (5.3.10). Similarly, when γ = 1, we obtain from (5.3.9) 1 2

G 00 (t) ≥ (2 + q) kut (t, ·)k22 + 2{E1 − Eu(0)} + α0 + 2qI (t) > (2 + q) {kut (t, ·)k22 + α0 } + 2qI (t).

Proceeding word by word as in the proof of Theorem 5.3.1 we obtain T0 given by (5.3.10). 2 Theorem 5.3.2 generalizes Theorem 2.4 of [29], where a negative initial energy, cooperative initial data and a linear external damping have been considered. In the next result we treat the limit case Eu(0) = E0 , under the hypothesis (D)0 introduced in Section 5.2. Before doing this observe that for the model (5.1.1) condition (D)0 –(i) simplifies considerably, since Q is linear in ut , so that hQ(t, ·, u, ut ), ut i = Q(t)kut (t, ·)k22 . Hence, (D)0 –(i) and (D)0 –(iii) can be combined together in the request that max{Q(0), %(0)} > 0. Note that (D)0 –(i) is automatic for problem (5.1.1) when Q(0) > 0, since Q satisfies (5.1.2).

142


Theorem 5.3.3. Suppose that either max{Q(0), %(0)} > 0 or (D)0 –(ii) holds. Let u ∈ X be a solution of (5.1.1), satisfying the limit condition (4.5.17), namely ku0 kq > υ0 , Eu(0) = E0 . Assume that

hu0 , u1 i >

 1  [Q(0)ku0 k22 + %(0)kDL u0 k22 ] ,   γ − 1

γ > 1,

    2 [Q(0)ku0 k22 + %(0)kDL u0 k22 ] , q−2

γ = 1,

(5.3.11)

and denote by  ku0 k22   , γ > 1,    (γ − 1)hu0 , u1 i − [Q(0)ku0 k22 + %(0)kDL u0 k22 ] (5.3.12) T0 =  2  2ku0 k2    , γ = 1. (q − 2)hu0 , u1 i − 2 [Q(0)ku0 k22 + %(0)kDL u0 k22 ] Then T ≤ T0 . Proof. Let u ∈ X be a solution of (5.1.1), satisfying (4.5.17). Proceeding as in the proof of Theorem 5.3.2, we first observe that Eu(0) = E0 < E1 < E2 by Lemma 4.5.6. Let us consider the function G of Theorem 5.3.1, replacing β with 0. In particular, when γ > 1 from (5.3.2) we now get 1 2

G 00 (t) ≥ (1 + γ) kut (t, ·)k22 + 2γ{E2 − Eu(0)} + 2γI (t) > (1 + γ) kut (t, ·)k22 + 2γI (t) ≥ 0,

where as always I is defined in (5.3.3). Hence also in the case β = 0 the main request (5.3.5) is satisfied by (5.3.11), being hu0 , u1 i > 0, so that ku0 k2 > 0 and ku1 k2 > 0 and in turn G (0) > 0. From now on the proof is as in Theorem 5.3.1, with T0 now given by (5.3.12). Similarly, when γ = 1, taking again β = 0 in the definition of G , we obtain from (5.3.9) 1 2

G 00 (t) ≥ (2 + q) kut (t, ·)k22 + 2{E1 − Eu(0)} + 2qI (t) > (2 + q) kut (t, ·)k22 + 2qI (t) ≥ 0.

143

Noting again that the main request (5.3.5) holds also in this setting by (5.3.11), we can proceed word by word as in the proof of Theorem 5.3.1 getting T0 as in (5.3.12). 2 It should be noted that when max{Q(0), %(0)} = 0 we have that Q ≡ 0 as well as % ≡ 0, by (5.1.2) and (5.1.3). In this case, if f does not depend on t, Theorem 5.3.3 cannot be applied, since assumption (D)0 –(ii) does not hold. However, as far as we know, Theorems 5.3.2 and 5.3.3 constitute the first attempt to provide a priori estimates for the lifespan of maximal solutions for models like (5.1.1), in such generality.

5.4

Blow up at Infinity for the Second Model

In this section we present our first blow up result for (5.1.4). Throughout the section we assume (F1 ), (F2 ) and (F4 )0 , with q satisfying (5.2.5), namely (F4 )0 There exists a number q such that ( ≤ 2∗L = 2n/(n − 2L), if n > 2L, 2γ < q (5.4.1) < ∞, if 1 ≤ n ≤ 2L, and for all F > 0 and φ ∈ X for which inf t∈R+0 F φ(t) ≥ F, there exist c1 = c1 (F, φ) > 0 and ε0 = ε0 (F, φ) > 0 such that (i)

F φ(t) ≤ c1 kφ(t, ·)kqq for all t ∈ R+ 0,

and for all ε ∈ (0, ε0 ] there exists c2 = c2 (F, φ, ε) > 0 such that (ii) hf (t, ·, φ(t, ·)), φ(t, ·)i − (q − ε)F φ(t) ≥ c2 kφ(t, ·)kqq for all t ∈ R+ 0.

Lemma 5.4.1. If u ∈ X is a solution of (5.1.4) with Eu(0) < E1 , then w1 > 0, w2 > 0 and E1 > 0. Proof. Let u ∈ X be a solution of (5.1.4) in R+ 0 × Ω, as in the statement. Clearly w1 ≥ 0 by (A ). Furthermore, by (E ), (B)–(ii) and the fact that Du is non-negative, F u(t) ≥ w1 − Eu(0) for all t ∈ R+ 0 , so that w2 ≥ w1 − Eu(0) > 2γw1 /q ≥ 0, being Eu(0) < E1 . Hence w2 > 0.

144


By (F3 ), in correspondence to F = w2 > 0, φ = u ∈ X, there exists ε0 = ε0 (w2 , u) > 0 and c1 = c1 (w2 , u) > 0 such that for all t ∈ R+ 0 ku(t, ·)kq ≥ c˜1 > 0 and kDL u(t, ·)k2 ≥ c˜1 /Sq ,

(5.4.2)

where c˜1 = (w2 /c1 )1/q > 0 and Sq = Sq (d, L, Ω) is the constant of the Sobolev embedding H0L (Ω) ,→ Lq (Ω). Moreover, by (5.1.5) we have 2(γ−1) 2A u(t) ≥ a + bkDL u(t, ·)k2 kDL u(t, ·)k22 ≥ a1 kDL u(t, ·)k22 , (5.4.3) where a1 = a + b(˜ c1 /Sq )2(γ−1) > 0. In particular, a1 w1 ≥ inf kDL u(t, ·)k22 > 0, 2 t∈R+0 2

as claimed, and in turn E1 > 0 by (5.4.1).

Furthermore we assume that Q verifies the following condition (Q)0 (Q)0 Along every solution u ∈ X of (5.1.4), there exist t ≥ 0, q1 > 0, m > 1, κ ≥ 0, ℘ > 1 with m + κ ≤ ℘ < q, non-negative functions δ1 , 1,1 δ2 ∈ L ∞ loc (J), and positive functions ψ, k ∈ Wloc (J), J = [t, ∞), with k 0 ≥ 0, such that for all t ∈ J 0 Q(t) ≤q1 δ1 (t)1/m ku(t, ·)kκ/m Du(t)1/m q 0 + δ2 (t)1/℘ Du(t)1/℘ ku(t, ·)kq , 1/(m−1)

1/(℘−1)

δ1 + δ2 ≤ k/ψ, ( o(ψ(t)), if k%k∞ < ∞, ψ 0 (t) = o(ψ(t)/%(t)), if k%k∞ = ∞,

(5.4.4) as t → ∞,

where Q(t) = hQ(t, ·, u(t, ·), ut (t, ·)), u(t, ·)i. Without loss of generality, taking t even larger if necessary, we assume that %(t) > 0. We refer to the end of this section for specific examples of functions Q verifying condition (Q)0 . For simplicity of notation let us introduce the negative numbers κ q q α1 = 1 + − , α2 = 1 − . (5.4.5) m m ℘ Actually, α1 ≤ α2 < 0, since (℘ − m)q ≥ κq ≥ κ℘, being κ ≥ 0, and ℘ < q in (Q)0 .

145

Theorem 5.4.2. Let u ∈ X be a solution of (5.1.4) such that Eu(0) < E1 .

(5.4.6)

If condition (Q)0 holds along u and for all c0 > 0 R t exp c0 t ψ(τ ) [max{k(τ ), ψ(τ )}]−1 dτ = ∞, if k%k∞ < ∞, lim t→∞ max{k(t), ψ(t)} R t −1 exp c0 t ψ(τ ) [max{k(τ ), %(τ )ψ(τ )}] dτ lim = ∞, if k%k∞ = ∞, t→∞ max{k(t), %(t)ψ(t)} (5.4.7) then lim ku(t, ·)kq = ∞. (5.4.8) t→∞

Proof. Let u ∈ X be a solution of (5.1.4) satisfying (5.4.6). Take φ = u in the Distribution Identity (A), thus o dn hut (t, ·),u(t, ·)i + %(t)K (kDL u(t, ·)k22 )/2 dt = kut (t, ·)k22 − hAu(t, ·), u(t, ·)i + hf (t, ·, u), u(t, ·)i (5.4.9) %0 (t) K (kDL u(t, ·)k22 ) − Q(t), + 2 where Q(t) = hQ(t, ·, u(t, ·), ut (t, ·)), u(t, ·)i as defined in (Q)0 . Since u satisfies (5.4.6), then w1 > 0, w2 > 0 and E1 > 0 by Lemma 5.4.1. Hence, in correspondence to F = w2 and φ = u, there exists ε0 = ε0 (w2 , u) > 0 such that inequalities (i) and (ii) of (F4 )0 hold true. Without loss of generality, we take ε0 > 0 so small that ε0 w1 ≤ (q − 2γ)w1 − q[Eu(0)]+ = q{E1 − [Eu(0)]+ },

(5.4.10)

which is possible by (5.4.6). We remark that (5.4.10) forces ε0 ≤ q − 2γ. Fix ε ∈ (0, ε0 ). By (5.2.1) and (F4 )0 –(ii) hAu(t, ·), u(t, ·)i − hf (t, ·, u(t, ·)), u(t, ·)i ≤ 2γA u(t) − (q − ε)F u(t) − c2 ku(t, ·)kqq .

146


Since q − 2γ > ε, then c3 = ε(q − ε − 2γ)/2q > 0. Moreover, by (E ) we have that F u(t) ≥ A u(t) − Eu(t), so that 2γA u(t) − (q − ε)F u(t) ≤ (q − ε)Eu(t) − (q − ε − 2γ)A u(t) q−ε ≤ (q − ε)Eu(0) −(q− ε−2γ) 1− A u(t) q q−ε − (q − ε − 2γ) w1 q ≤ (q − ε){[Eu(0)]+ − E1 + εw1 /q} − 2c3 A u(t) < −c3 M (kDL u(t, ·)k22 ), by (5.4.6), (5.4.10) and (A ). Combining the last inequalities together with (5.4.9) we get o dn hut (t, ·), u(t, ·)i + %(t)K (kDL u(t, ·)k22 )/2 (5.4.11) dt 2 q 2 ≥ kut (t, ·)k2 + c2 ku(t, ·)kq + c3 M (kDL u(t, ·)k2 ) − Q(t), where we have also used the non-negativity of %0 and K . Let us estimate the main dynamic quantity Q(t) using (Q)0 . By (5.4.4) 1/m0 Q(t) ≤ q1 δ1 (t)1/(m−1)Du(t) ku(t, ·)k1+κ/m q o 0 1/℘ + δ2 (t)1/(℘−1) Du(t) ku(t, ·)kq ( h i 1/m0 δ1 (t)1/(m−1)Du(t) = q1 ku(t, ·)kq/m ku(t, ·)kαq 1 q n

+ (" ≤ q1

h

δ2 (t)1/(℘−1) Du(t)

1/℘0

i

)

ku(t, ·)kq/℘ ku(t, ·)kαq 2 q

# 1/(m−1) 2 ` δ1 (t) Du(t) + ku(t, ·)kqq ku(t, ·)kαq 1 ` 2 # ) " 1/(℘−1) ` 2 δ2 (t) + Du(t) + ku(t, ·)kqq ku(t, ·)kαq 2 , ` 2

147

for all t ∈ J, where in the last step we have applied Young’s inequality, with ` > 0 to be fixed later. Consequently, being α1 ≤ α2 < 0, it follows n Q(t) ≤ q2 `˜ δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) Du(t) o (5.4.12) + `ku(t, ·)kqq ku(t, ·)kαq 2 n o ˜ ≤ q3 `k(t)Du(t)/ψ(t) + `ku(t, ·)kqq , 0 0 with q2 = q1 max{1, (˜ c1 )α1 −α2 }, `˜ = max{(2/`)m /m , (2/`)℘ /℘ }, q3 = q2 c˜α1 2 > 0 and in the last step we have used (5.4.4)2 . Introduce the function Z t H (t) = H0 + Du(τ )dτ (5.4.13)

0

for all t ∈ R+ 0 , where H0 is any number in the interval (0, E1 − Eu(0)]. Of course, H is well defined by (B)–(i) and non-decreasing, being D = H 0 ≥ 0. Define for all t ∈ R+ 0 the main auxiliary function n o Z (t) = λk(t)H (t) + ψ(t) hut , ui + %(t)K (kDL u(t, ·)k22 )/2 , (5.4.14) 1,1 where λ > 0 is a constant to be fixed later. Clearly Z ∈ Wloc (J) by Corollary 8.10 of [23] and a.e. in J, n o Z 0 =λk 0 H + λkH 0 + ψ 0 hut , ui + %(t)K (kDL uk22 )/2 (5.4.15) o dn +ψ hut , ui + %(t)K (kDL uk22 )/2 . dt

Hence, putting (5.4.11) and (5.4.12) into (5.4.15), and recalling that Du = H 0 , we get n o 0 0 0 0 2 Z ≥ λk H + λkH + ψ hut , ui + %(t)K (kDL uk2 )/2 n o + ψ kut (t, ·)k22 + c2 ku(t, ·)kqq + c3 M (kDL u(t, ·)k22 ) − Q(t) n o 0 0 2 ˜ ≥ k λ − q3 ` H + ψ hut , ui + %(t)K (kDL uk2 )/2 o n + ψ kut k22 + (c2 − q3 `)kukqq + c3 M (kDL uk22 ) ,

148


where in the last step we have also used the fact that λk 0 H ≥ 0. Next, by the Cauchy and Young inequalities, and the definition of X, we have |hu(t, ·), ut (t, ·)i| ≤ kut (t, ·)k2 ku(t, ·)k2 ≤ kut (t, ·)k22 + ku(t, ·)k22 . The embedding Lq (Ω) ,→ L2 (Ω) is continuous by (5.4.1), so that ku(t, ·)k2 ≤ |Ω|(q−2)/2q ku(t, ·)kq , and by (5.4.2), this gives ku(t, ·)k22 ≤ |Ω|1−2/q ku(t, ·)k2q ≤ c4 ku(t, ·)kqq , where c4 = |Ω|1/q /˜ c1

q−2

, being q > 2γ ≥ 2. Hence

|hu(t, ·), ut (t, ·)i| ≤ kut (t, ·)k22 + c4 ku(t, ·)kqq .

(5.4.16)

Clearly −ψ 0 ≥ −|ψ 0 |, and so a.e. in J, by (K ) n Z 0 ≥k λ − q3 `˜ H 0 + ψ (1 − |ψ 0 |/ψ) kut k22 + (c2 − q3 ` − c4 |ψ 0 |/ψ) kukqq o + (c3 cσ − %(t)|ψ 0 |/2ψ) K (kDL uk22 ) . Fix ` > 0 so small that 2q3 ` < c2 and consider a time T1 in the interval J enough large to have 2|ψ 0 |/ψ ≤ min{1, (c2 − q3 `)/c4 } and %|ψ 0 |/ψ ≤ c3 cσ in J1 = [T1 , ∞), since ψ 0 (t) = o(ψ(t)) and %(t)ψ 0 (t) = o(ψ(t)) as t → ∞, by (5.4.4)3 , both when k%k∞ < ∞ and k%k∞ = ∞. Then take λ > 0 so large that λ ≥ q3 `˜ and Z (T1 ) > 0. In conclusion, we have shown that Z 0 (t) ≥ B 0 ψ(t) kut (t, ·)k22 +ku(t, ·)kqq +K (kDL u(t, ·)k22 ) ≥ 0, (5.4.17) for a.a. t ∈ J1 , where 2B 0 = min{1, c2 − q3 `, c3 cσ }. Hence, Z (t) ≥ Z (T1 ) > 0 for all t ∈ J1 , being Z (T1 ) > 0. Now observe that, by (B)–(ii), (E ), (F4 )0 –(i) and the choice of H0 , for all t ∈ R+ 0 we have H (t) ≤ H0 + Eu(0) − Eu(t) ≤ H0 + Eu(0) − A u(t) + F u(t) < F u(t) ≤ c1 ku(t, ·)kqq , (5.4.18)

149

since A u(t) ≥ w1 > E1 by (5.4.1). Consequently, by (5.4.14), (5.4.16), (5.4.18) and (5.1.7) it follows that for all t ∈ J1 n Z (t) ≤ λc1 k(t)ku(t, ·)kqq + ψ(t) kut (t, ·)k22 + c4 ku(t, ·)kqq o 2 + %(t)K (kDL u(t, ·)k2 )/2 kut (t, ·)k22 c4 ku(t, ·)kqq q ≤ λc1 k(t)ku(t, ·)kq +%(t)ψ(t) + %(T1 ) %(T1 ) 2 K (kDL u(t, ·)k2 ) , + 2 so that, putting B = max{1/%(T1 ), λc1 + c4 /%(T1 ), 1/2}, we get n Z (t) ≤B max{k(t), %(t)ψ(t)} · kut (t, ·)k22 o (5.4.19) + ku(t, ·)kqq + K (kDL u(t, ·)k22 ) . Therefore, combining (5.4.17) with (5.4.19), we immediately obtain Z −1 Z 0 ≥ c0 ψ[max{k, %ψ}]−1 ,

(5.4.20)

where c0 = B 0 /B. On the other hand, by (B)–(ii), (E ), (5.4.1) and the definition of w1 , for all t ∈ R+ 0 H0 ≤ H (t) ≤ H0 + Eu(0) − Eu(t) ≤ E1 − 21 kut (t, ·)k22 + M (kDL u(t, ·)k22 ) + F u(t). In particular, kut (t, ·)k22 + M (kDL u(t, ·)k22 ) ≤ 2[E1 − H0 + F u(t)] ≤ 2[E1 + F u(t)], and in turn, using (F4 )0 –(i), we get kut (t, ·)k22 +K (kDL u(t, ·)k22 ) ≤ Cσ [E1 +F u(t)] ≤ CK ku(t, ·)kqq , (5.4.21) where Cσ = 2 max{1, c−1 σ }, cσ is the constant given in (K ) corresponding to σ = (˜ c1 /Sq )2 determined in (5.4.2), and finally CK = Cσ max{E1 /˜ c1 , c1 }. Hence, putting (5.4.21) into (5.4.19), we get for all t ∈ J1 Z (t) ≤ C max{k(t), %(t)ψ(t)}ku(t, ·)kqq

for all t ∈ J1 ,

(5.4.22)

150


where C = B(1 + CK ). Case k%k∞ < ∞. Relations (5.4.20) and (5.4.22) simplify. In particular, t ∈ J1 Z t −1 Z (t) ≥ Z (T1 ) exp c% ψ(τ ) [max{k(τ ), ψ(τ )}] dτ , (5.4.23) T1

where c% = c0 / max{1, k%k∞ }. Therefore, by (5.4.22), for all t ≥ T1 R t −1 exp c% T1 ψ(τ ) [max{k(τ ), ψ(τ )}] dτ , (5.4.24) ku(t, ·)kqq ≥ Z0 · max{k(t), ψ(t)} where Z0 = Z (T1 )/C% > 0 and C% = C max{1, k%k∞ }. Thus for all t ∈ J1 ⊂ J R t exp c% T1 ψ(τ ) [max{k(τ ), ψ(τ )}]−1 dτ max{k(t), ψ(t)} R t −1 exp c% t ψ(τ ) [max{k(τ ), ψ(τ )}] dτ , = E0 max{k(t), ψ(t)} RT where E0 = exp −c% t 1 ψ(τ ) [max{k(τ ), ψ(τ )}]−1 dτ > 0. Therefore, passing to the limit as t → ∞ in (5.4.24), from (5.4.7), valid in particular for c0 = c% > 0, we obtain (5.4.8), and the proof is complete when k%k∞ < ∞. Case k%k∞ = ∞. Without loss of generality we can suppose % ≥ 1 in J1 , assuming T1 even larger if necessary, being limt→∞ %(t) = k%k∞ = ∞ by (5.1.7). Hence, by (5.4.20) Z t −1 Z (t) ≥ Z (T1 ) exp c0 ψ(τ ) [max{k(τ ), %(τ )ψ(τ )}] dτ . (5.4.25) T1

Combining (5.4.25) with (5.4.22), we get for all t ≥ T1 R t −1 exp c0 T1 ψ(τ ) [max{k(τ ), %(τ )ψ(τ )}] dτ ku(t, ·)kqq ≥ Z0 · , max{k(t), %(t)ψ(t)} where Z0 = Z (T1 )/C > 0. Property (5.4.8) follows at once by (5.4.7)2 . 2

151

Remark 5.4.3. In Theorem 5.4.2 we have %(t)K(kDL u(t, ·)k22 ) > 0 for t ≥ t by (5.1.6), (5.1.7) and (5.4.2). We point out that when %(t)K(kDL u(t, ·)k22 ) ≡ 0 in R+ 0 , global solutions of (5.1.4) do not exist when Eu(0) is bounded above by the value E2 given in Figure 4.1, as shown in Theorem 3.1 of [9]. In this final part of the section we give some applications of Theorem 5.4.2, in special subcases of the external damping and the source force. We start by giving a prototype for Q, satisfying (Q)0 and (5.4.7). Let us consider a continuous function Q verifying inequality (4.1.2). Moreover, assume that there exists t >> 1 such that Q(t, x, u, v) = d1 (t, x)|u|κ |v|m−2 v + d2 (t, x, u)|v|℘−2 v m > 1 κ ≥ 0 ℘ > 1, with m + κ ≤ ℘ < q

(5.4.26)

for all (t, x, u, v) ∈ J × Ω × Rd × Rd , where J = [t, ∞), the functions + ℘1 ∞ d d1 ∈ C(R+ 0 → L (Ω)), d2 ∈ C(R0 → L (Ω × R )) are non-negative and ℘1 = q/(q − κ − m). Define δ1 (t) = kd1 (t, ·)k℘1 and δ2 (t) = |Ω|(q−℘)/q sup(x,ξ)∈Ω×Rd d2 (t, x, ξ). Without loss of generality, we take t so large that %(t) > 0. Proposition 5.4.4. Let Q be of the type given in (5.4.26), then along any solution u of (5.1.4) n 0 |Q(t)| ≤ δ1 (t)1/m ku(t, ·)kκ/m Du(t)1/m q o (5.4.27) 0 +δ2 (t)1/℘ Du(t)1/℘ ku(t, ·)kq for all t ∈ J. Moreover, (Q)0 and (5.4.7) are verified if f satisfies (4.1.3), (F1 ) and (F2 ) and one of the following conditions holds (i) k%k∞ < ∞ and for each t ≥ t δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) ≤ K(1 + t)s/(m−1) , with K ≥ 1 and 0 ≤ s ≤ m − 1;

(5.4.28)

152


(ii) k%k∞ = ∞, for each t ≥ t δ1 (t)1/(m−1) + δ2 (t)1/(℘−1) ≤ Const. %(t),

(5.4.29)

and for all c0 > 0 Z t 1 dτ lim exp c0 = ∞. t→∞ %(t) t %(τ )

(5.4.30)

Proof. The proof of (5.4.27) is analogous to the proof of Proposition 4.4.1, with q ≡ Const. Case (i). It is enough to take k(t) = K(1+t)s/(m−1) , K ≥ 1 and ψ(t) = 1 for all t ∈ J, if 0 ≤ s < m − 1; while k(t) = K and ψ(t) = (1 + t)−1 , 1,1 (J) are for all t ∈ J, if s = m − 1. In both the situations ψ, k ∈ Wloc 0 0 positive, k ≥ 0, ψ (t) = o(ψ(t)) as t → ∞, k ≥ ψ in J, being K ≥ 1, (5.4.4)2 is verified in J and (5.4.7)1 holds. For a proof of these facts we refer to [9, Proposition 4.1], as well as to the proof of Proposition 4.4.1, with q ≡ Const. Case (ii). Put k(t) = Const. %(t) , with Const. ≥ 1 and ψ(t) = 1 1,1 for all t ∈ J, so that k ≥ %ψ. In this case ψ, k ∈ Wloc (J) are positive, k 0 ≥ 0, ψ 0 (t) = o(ψ(t)/%(t)) as t → ∞, (5.4.4)2 is verified in J by (5.4.29) and (5.4.7)2 holds by (5.4.30). 2 For the next theorem we require that Q satisfies all the assumptions in Proposition 5.4.4 and that f is of the type given in (5.2.5)–(5.2.6). Hence, by Proposition 5.2.1, conditions (F1 ), (F2 ) and (F4 )0 –(i) are satisfied. Moreover we assume that f satisfies (5.2.14), that is 1≤σ 0,

so that also (F4 )0 –(ii) is verified with ε0 ∈ (0, q − σ] and c2 = cε/q for all ε ∈ (0, ε0 ], as proved at the end of Section 5.2. Hence, all the structural assumptions of Theorem 5.4.2 are fulfilled. Theorem 5.4.5. Let u ∈ X be a solution of (5.1.4) such that one of the following conditions holds (i) υ(0) > υ0 and Eu(0) < E0 , (ii) υ(0) > υ0 , Eu(0) = E0 and (D)0 .

153

Then limt→∞ ku(t, ·)kq = ∞. Proof. Let u ∈ X be a solution of (5.1.4) as in the statement. If (i) holds, then by Lemma 4.5.4 we have that Eu(0) < E0 < E1 . On the other hand, if (ii) holds, then Eu(0) < E1 by Lemma 5.2.3. In both cases the claim follows directly applying Theorem 5.4.2. 2 Remark 5.4.6. In Theorem 5.4.5 conditions % > 0 definitively and K not trivial are crucial. Indeed, if %(t)K(kDL u(t, ·)k22 ) ≡ 0 in R+ 0 , global solutions may not exist by Theorem 5.1 of [9]. + Consider now Q(t, x, u, v) = d1 (t)v, where d1 ∈ C 1 (R+ 0 → R0 ) and 0 d1 ≤ 0. Clearly Q satisfies (5.4.26), with κ = 0, m = 2, d1 (t, x) = d1 (t) and d2 ≡ 0. Assume furthermore that % ≡ Const. > 0 and K(τ ) ≡ 1. In this situation, Theorems 7.2 and 7.3 of [9] say that there are no solutions u of (5.1.4) defined in the whole space R+ 0 × Ω, when, in our notation, Eu(0) ≤ E0 , condition (D)0 holds in the limit case Eu(0) = E0 and hu(0, ·), ut (0, ·)i is bounded from below by an appropriate constant, depending only on the initial data of the problem. This means that Theorem 5.4.5 can be applied only when hu(0, ·), ut (0, ·)i is sufficiently small. In [87] Ono considered a subcase of (5.1.4), with Q ≡ 0, % ≡ 1 and K ≡ 1. For this model, in Theorem 3 of [87], he proved the existence of a global solution u, assuming Eu(0) limited above and υ(0) = ku(0, ·)kq small enough. Furthermore he showed that υ(t) = ku(t, ·)kq → 0 as t → ∞. Under the assumptions of Theorem 5.4.5 of this paper, condition υ(0) > υ0 > 0 actually implies υ(t) > υ0 for all t ∈ R+ 0.

Bibliography [1] R.A. Adams, Sobolev spaces, Pure and Applied Mathematics, 65. Academic Press, New York-London, 1975, xviii+268 pp. [2] N. Alaa, M. Guedda, Nonexistence of global solutions to a class of nonlinear differential inequalities and application to hyperbolic– elliptic problems, J. Appl. Anal., 9 (2003), 103–121. [3] M.J. Alves, R.B. Assun¸caõ, P.C. Carrião, O.H. Miyagaki, Multiplicity of nontrivial solutions to a problem involving the weighted p-biharmonic operator, Mat. Contemp., 36 (2009), 11–27. [4] C.O. Alves, F.J.S.A. Corrêa, T.F. Ma, Positive Solutions for a Quasilinear Elliptic Equation of Kirchhoff Type, Comput. Math. Appl., 49 (2005), 85–93. [5] A. Ambrosetti, P.H. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications, J. Funct. Anal., 14 (1973), 349– 381. [6] S. Antontsev, S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness, localization properties of solutions, Nonlinear Anal., 65 (2006), 728–761. [7] D. Arcoya, J. Carmona, A nondifferentiable extension of a theorem of Pucci and Serrin and applications, J. Differential Equations, 235 (2007), 683–700. [8] G. Autuori, Kirchhoff Systems: Asymptotic stability, Global Nonexistence and Blow up, Ph.D. Thesis, University of Florence, 2010.

156

Bibliography

[9] G. Autuori, F. Colasuonno, P. Pucci, Lifespan estimates for solutions of polyharmonic Kirchhoff Systems, in press in Math. Models Methods Appl. Sci., 22 (2012) DOI: 10.1142/S0218202511500096. [10] G. Autuori, F. Colasuonno, P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Var. Elliptic Equ., 57 (2012), 379–395. [11] G. Autuori, P. Pucci, Kirchhoff systems with nonlinear source and boundary damping terms, Commun. Pure Appl. Anal., 9 (2010), 1161–1188. [12] G. Autuori, P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952–1965. [13] G. Autuori, P. Pucci, Local asymptotic stability for polyharmonic Kirchhoff systems, Appl. Anal., 90 (2011), 493–514. [14] G. Autuori, P. Pucci, M.C. Salvatori, Asymptotic Stability for Nonlinear Kirchhoff Systems, Nonlinear Anal. Real World Appl., 10 (2009), 889–909. [15] G. Autuori, P. Pucci, M.C. Salvatori, Global Nonexistence for Nonlinear Kirchhoff Systems, Arch. Ration. Mech. Anal., 196 (2010), 489–516. [16] J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser 2, 28 (1977), 473– 486. [17] E. Bcache, G. Derveaux, P. Joly, An efficient numerical method for the resolution of the Kirchhoff-Love dynamic plate equation, Numer. Methods Partial Differential Equations, 21 (2005), 323–348. [18] M.S. Berger, Nonlinearity and functional analysis, Lectures on Nonlinear Problems in Mathematical Analysis, Pure and Applied Mathematics, Academic Press, New York–London, 1977, xix+417 pp. [19] A. Boccuto, E. Vitillaro, Asymptotic stability for abstract evolution equations and applications to partial differential systems, Rend. Circ. Mat. Palermo, 47 (1998), 25–48.

Bibliography

157

[20] G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal., 54 (2003), 651–665. [21] M.M. Boureanu, P. Pucci, V.D. R˘adulescu, Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent, Complex Var. Elliptic Equ., 56 (2011), 755–767. [22] B.E. Breckner, D. Repovˇs, Cs. Varga, On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket, Nonlinear Anal., 73 (2010), 2980–2990. [23] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011, xiv+599 pp. [24] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55–64. [25] F. Cammaroto, L. Vilasi, Multiple solutions for a Kirchhoff–type problem involving the p(x)–Laplacian operator, Nonlinear Anal., 74 (2011), 1841–1852. [26] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705–731. [27] M.M. Cavalcanti, V.N. Domingos Cavalcanti, P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119–158. [28] C. Chen, H. Wang, Ground state solutions for singular p–Laplacian equation in RN , J. Math. Anal. Appl., 351 (2009), 773–780. [29] W. Chen, Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal., 70 (2009), 3203–3208. [30] P.G. Ciarlet, Mathematical elasticity. Vol. I. Three-dimensional elasticity Studies in Mathematics and its Applications 20, NorthHolland Publishing Co., Amsterdam, 1988, xlii+451 pp.

158

Bibliography

[31] P.G. Ciarlet, P. Destuynder, A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg., 17/18 (1979), 227–258. [32] F. Colasuonno, P. Pucci, Multiplicity of solutions for p(x)– polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974. [33] F. Colasuonno, P. Pucci, Cs. Varga, Multiple solutions for an eigenvalue problem involving p–Laplacian type operators, in press in Nonlinear Anal. (2011), DOI:10.1016/j.na.2011.09.048. [34] F. Colin, Existence result for a class of nonlinear elliptic systems on punctured unbounded domains, Bound. Value Probl. 2010, Art. ID 175409, 15 pp. [35] F.J.S.A. Corrêa, G.M. Figueiredo, On an elliptic equation of p– Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 236–277. [36] F.J.S.A. Corrêa, S.D.B. Menezes, Existence of solutions to nonlocal and singular elliptic problems via Galerkin method, Electron. J. Differential Equations, 2004 (2004), 1-10. [37] M. Cuesta, Eigenvalue problems for the p–Laplacian with indefinite weights, Electron. J. Differential Equations, 2001 (2001), 1–9. [38] G. Dai, R. Hao, Existence of solutions for a p(x)–Kirchhoff–type equation, J. Math. Anal. Appl., 359 (2009), 275–284. [39] G. Dai, J. Wei, Infinitely many non–negative solutions for a p(x)– Kirchhoff–type problem with Dirichlet boundary condition, Nonlinear Anal., 73 (2010), 3420–3430. [40] P. D’Ancona, Y. Shibata, On global solvability of non–linear viscoelastic equations in the analytic category, Math. Meth. in Appl. Sciences, 17 (1994), 477–486. [41] M. Dauge, E. Faou, Koiter estimate revisited, Math. Models Methods Appl. Sci., 20 (2010), 1–42.

Bibliography

159

[42] F. Demengel, E. Hebey, On some nonlinear equations involving the p–Laplacian with critical Sobolev growth, Adv. Differential Equations, 3 (1998), 533-574. [43] P.L. De Nápoli, M.C. Mariani, Mountain pass solutions to equations of p–Laplacian type, Nonlinear Anal., 54 (2003), 1205–1219. [44] Y. Deng Y, H. Pi, Multiple solutions for p–harmonic type equations, Nonlinear Anal., 71 (2009), 4952–4959. [45] L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and W k,p(·) , Math. Nachr., 268 (2004), 31–43. [46] L. Diening, P. Harjulehto, P. Hästö, M. R˚ uˇziˇcka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes, 2017, Springer–Verlag, Berlin, 2011, ix+509 pp. [47] L. Diening, M. R˚ uˇziˇcka, Calderón–Zygmund operators on generalized Lebesgue spaces Lp(·) and problems related to fluid dynamics, J. Reine Angew. Math., 563 (2003), 197–220. ˆ [48] P. Drábek, M. Otani, Global bifurcation result for the p–biharmonic operator, Electron. J. Differential Equations, 2001 (2001), 1–19. [49] D.E. Edmunds, J. Rákosn´ık, Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267–293. [50] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)–Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843–1852. [51] X. Fan, Q. Zhang, D. Zhao Eigenvalues of p(x)–Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005),306-317. [52] X.L. Fan, D. Zhao, On the spaces Lp(x) and W m,p(x) , J. Math. Anal. Appl., 263 (2001), 424–446. [53] F. Gazzola, H.–C. Grunau, G. Sweers, Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics, 1991, Springer-Verlag, Berlin, 2010, xviii+423 pp.

160

Bibliography

[54] A.N. Gent, Engineering with Rubber: How to Design Rubber Components, 2nd ed. H. Gardner Publ., 2001, 365 pp. [55] S. Gerbi, B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), 1051–1074. [56] V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295–308. [57] G.C. Gorain, Exponential energy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation, Appl. Math. Comput., 177 (2006), 235–242. [58] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, Boston, 1985, xiv+410 pp. [59] M. Guedda, H. Labani, Nonexistence of global solutions to a class of nonlinear wave equations with dynamic boundary conditions, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 39–46. [60] P. Harjulehto, P. Hästö, M. Koskenoja, S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal., 25 (2006), 205–222. [61] P. Harjulehto, P. Hästö, U.V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. [62] A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications, Nonlinear Anal., 72 (2010), 1319-1338. [63] Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and some Applications, Cambridge University Press, Cambridge, 2003, xii+369 pp. [64] A. El Khalil, S. Kellati, A. Touzani On the spectrum of the pbiharmonic operator, Electron. J. Differ. Equ. Conf., 9 (2002), 161– 170.

Bibliography

161

[65] G. Kirchhoff, Vorlesungen u ¨ber mathematische Physik: Mechanik, Teubner, Leipzig, 1883. [66] S. Koumou Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 299–314. [67] O. Kováˇcik, J. Rákosn´ık, On spaces Lp(x) and W 1,p(x) , Czechoslovak Math. J., 41 (1991), 592–618. [68] A. Kratochv´ıl, J. Nec˘as, The discreteness of the spectrum of a nonlinear Sturm-Liouville equation of fourth order, Comment. Math. Univ. Carolinæ, 12 (1971), 639-653 (in Russian). [69] A. Kristály, Cs. Varga, Multiple solutions for elliptic problems with singular and sublinear potentials, Proc. Amer. Math. Soc., 135 (2007), 2121–2126. [70] A. Kristály, H. Lisei, Cs. Varga, Multiple solutions for p–Laplacian type equations, Nonlinear Anal., 68 (2008), 1375–1381. [71] A. Kristály, V. R˘adulescu, Cs. Varga, Variational principles in mathematical physics, geometry, and economics, Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010. [72] H.A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form P utt = Au + F (u), Trans. Amer. Math. Soc., 192 (1974), 138–146. [73] H.A. Levine, P. Pucci, J. Serrin, Some remarks on global nonexistence for nonautonomous abstract evolution equations, Contemp. Math., 208 (1997), 253-263. [74] H.A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341–361. [75] C. Li, C.L. Tang, Three solutions for a class of quasilinear elliptic systems involving the (p, q)–Laplacian, Nonlinear Anal., 69 (2008), 3322–3329.

162

Bibliography

[76] P. Lindqvist, On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Amer. Math. Soc., 109 (1990), 157–164. [77] J.L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), pp. 284– 346, North-Holland Math. Stud., 30, North-Holland, AmsterdamNew York, 1978. [78] A.E.H. Love, A treatise on the Mathematical Theory of Elasticity Fourth Edition, Dover Publications, New York, 1944. [79] V.F. Lubyshev, Multiple solutions of an even–order nonlinear problem with convex–concave nonlinearity, Nonlinear Anal., 74 (2011), 1345–1354. [80] T.F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type, Discrete Contin. Dyn. Syst., suppl. (2007), 694–703. [81] T.F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), e1967–e1977. [82] T.F. Ma, Positive solutions for a nonlinear Kirchhoff type beam equation, Appl. Math. Lett., 18 (2005), 479–482. [83] S.A. Messaoudi, B. Said Houari, A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation, Appl. Math. Lett., 20 (2007), 866–871. [84] Y. Mizuta, T. Ohno, T. Shimomura, N. Shioji, Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)–Laplacian and its critical exponent, Ann. Acad. Sci. Fenn. Math., 35 (2010), 115-130. [85] R. Monneau, Justification of the nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method, Arch. Ration. Mech. Anal., 169 (2003), 1–34. [86] R. Monneau, Some remarks on the asymptotic invertibility of the linearized operator of nonlinear elasticity in the context of the dis-

Bibliography

163

placement approach, ZAMM Z. Angew. Math. Mech., 86 (2006), 400-409.. [87] K. Ono, On Global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151–177. [88] K. Ono, Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation, Nonlinear Anal., 30 (1997), 4449–4457. [89] K. Ono, On the blowup problem for nonlinear Kirchhoff equations with nonlinear dissipative terms, J. Math. Tokushima Univ., 34 (2000), 1–8. [90] K. Ono, Global existence, asymptotic behaviour, and global nonexistence of solutions for damped non-linear wave equations of Kirchhoff type in the whole space, Math. Methods Appl. Sci., 23 (2000), 535–560. [91] K. Pfl¨ uger, Existence and multiplicity of solutions to a p–Laplacian equation with nonlinear boundary condition, Electron. J. Differential Equations, 10 (1998), 13 pp. (electronic). [92] D.R. Pitts, M.A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations, Indiana Univ. Math. J., 51 (2002), 1479–1509. [93] P. Pucci, J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. [94] P. Pucci, J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. [95] P. Pucci, J. Serrin, Asymptotic stability for non–autonomous dissipative wave systems, Comm. Pure Appl. Math., 49 (1996), 177–216. [96] P. Pucci, J. Serrin, Local asymptotic stability for dissipative wave systems, Israel J. Math., 104 (1998), 29–50.

164

Bibliography

[97] P. Pucci, J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203–214. [98] P. Pucci, R. Servadei, Existence, nonexistence and regularity of radial ground states for p–Laplacian equations with singular weights, Ann. Inst. H. Poincaré A.N.L., 25 (2008), 505–537. [99] B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems. Nonlinear operator theory, Math. Comput. Modelling, 32 (2000), 1485–1494. [100] B. Ricceri, On a three critical points theorem, Arch. Math., 75 (2000), 220–226. [101] B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084–3089. [102] B. Ricceri, A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151–4157. [103] P. Villaggio, Mathematical Models for Elastic Structures, Cambridge University Press, Cambridge, 1997, xii+679 pp. [104] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155–182. [105] E. Vitillaro, Some new results on global nonexistence and blowup for evolution problems with positive initial energy, Workshop on Blow–up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999), Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245–275. [106] W. Wang, P. Zhao, Nonuniformly nonlinear elliptic equations of p–biharmonic type, J. Math. Anal. Appl., 348 (2008), 730–738. [107] S.T. Wu, L.Y. Tsai, Blow–up of solutions for some non–linear wave equations of kirchhoff type with some dissipation, Nonlinear Anal., 65 (2006), 243–264.

Bibliography

165

[108] S.T. Wu, L.Y. Tsai, On a system of nonlinear wave equations of Kirchhoff type with a strong dissipation, Tamkang J. Math., 38 (2007), 1–20. [109] S.T. Wu, L.Y. Tsai, Existence and nonexistence of global solutions for a nonlinear wave equation, Taiwanese J. Math., 13 (2009), 2069– 2091. [110] Z. Yang, D. Qiu, Energy decaying and blow–up of solution for a Kirchhoff equation with strong damping, J. Math. Res. Exposition, 29 (2009), 707–715. [111] E. Zeidler, Nonlinear functional analysis and its applications, Vol. III, Variational methods and optimization, Springer–Verlag, New York, 1985, xxii+662 pp. [112] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343–358.