Bifurcation and multiplicity results for critical nonlocal

Bull. Sci. math. 140 (2016) 14–35

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Bulletin des Sciences Mathématiques www.elsevier.com/locate/bulsci

Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems ✩ Alessio Fiscella a , Giovanni Molica Bisci b,∗ , Raffaella Servadei c a

Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda 651, Campinas, SP CEP 13083-859 Brazil b Dipartimento PAU, Università ‘Mediterranea’ di Reggio Calabria, Via Melissari 24, 89124 Reggio Calabria, Italy c Dipartimento di Scienze di Base e Fondamenti, Università degli Studi di Urbino ‘Carlo Bo’, Piazza della Repubblica 13, 61029 Urbino, Pesaro e Urbino, Italy

a r t i c l e

i n f o

Article history: Received 13 May 2015 Available online 19 October 2015 MSC: primary 49J35, 35A15, 35S15 secondary 47G20, 45G05 Keywords: Fractional Laplacian Critical nonlinearities Best fractional critical Sobolev constant Variational techniques Integrodifferential operators

a b s t r a c t In this paper we consider the following critical nonlocal problem

∗

−LK u = λu + |u|2 u=0

−2

u

in Ω in Rn \ Ω ,

where s ∈ (0, 1), Ω is an open bounded subset of Rn , n > 2s, with continuous boundary, λ is a positive real parameter, 2∗ := 2n/(n − 2s) is the fractional critical Sobolev exponent, while LK is the nonlocal integrodifferential operator LK u(x) := x ∈ Rn ,

Rn

u(x + y) + u(x − y) − 2u(x) K(y) dy ,

whose model is given by the fractional Laplacian −(−Δ)s . ✩ The first author was supported by Coordenação de Aperfeiçonamento de pessoal de nível superior (CAPES) through the fellowship 33003017003P5–PNPD20131750–UNICAMP/MATEMÁTICA. The second and the third author were supported by the INdAM-GNAMPA Project 2015 Modelli ed equazioni non-locali di tipo frazionario. The first and the third author were supported by the FP7-IDEAS-ERC Starting Grant 2011 #277749 EPSILON (Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities) and the third author was supported by the MIUR National Research Project Variational and Topological Methods in the Study of Nonlinear Phenomena. * Corresponding author. E-mail addresses: fi[email protected] (A. Fiscella), [email protected] (G. Molica Bisci), raff[email protected] (R. Servadei).

http://dx.doi.org/10.1016/j.bulsci.2015.10.001 0007-4497/© 2015 Elsevier Masson SAS. All rights reserved.

A. Fiscella et al. / Bull. Sci. math. 140 (2016) 14–35

15

Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of −LK (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators. © 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction In recent years, nonlocal problems and operators have been widely studied in the literature and have attracted the attention of lot of mathematicians coming from different research areas. The interest towards equations involving nonlocal operators has grown more and more, thanks to their intriguing analytical structure and in view of several applications in a wide range of contexts. Indeed, fractional and nonlocal operators appear in concrete applications in many fields such as, among the others, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, thin obstacle problem, optimal transport, image reconstruction, through a new and fascinating scientific approach (see, e.g., the papers [2,8,11,12,18,23,37–39] and the references therein). After the seminal paper [10] by Brezis and Nirenberg, the critical problem

−Δu = λu + |u|2∗ −2 u in Ω u=0 on ∂Ω ,

(1.1)

has been widely studied in the literature (see, e.g. [13–17,19,20,22,26,35,40] just to name a few), also due to its relevant relations with problems arising in differential geometry and in physics, where a lack of compactness occurs (see, for instance, the famous Yamabe problem). Here Ω is an open bounded subset of Rn , n > 2, and 2∗ := 2n/(n − 2) is the critical Sobolev exponent. The first multiplicity result for problem (1.1) was proved by Cerami, Fortunato and Struwe in [14], where it was shown that in a suitable left neighborhood of any eigenvalue of −Δ (with Dirichlet boundary data) the number of solutions is at least twice the multiplicity of the eigenvalue. The authors also gave an estimate of the length of this neighborhood, which depends on the best critical Sobolev constant, on the Lebesgue measure of the set where the problem is set and on the space dimension. Later, in [15] the authors proved that in dimension n 6 and for λ > 0 less than the first eigenvalue of −Δ (with homogeneous Dirichlet boundary conditions), problem (1.1)

16


has at least two nontrivial solutions, one of which is a changing sign solution (for other results on changing sign solutions see, for instance, [17,26,35]). More recently, in [20] the authors improved the result got in [15], while in [19] Devillanova and Solimini proved the existence of infinitely many solutions for (1.1), provided the dimension n 7 and the parameter λ is positive. Finally, in [17] the authors showed that for n 4 problem (1.1) has at least (n + 1)/2 pairs of nontrivial solutions, provided λ > 0 is not an eigenvalue of −Δ, and (n + 1 − m)/2 pairs of nontrivial solutions, if λ is an eigenvalue of −Δ with multiplicity m < n + 2. When m n + 2, [17] gave no information about the multiplicity of solutions for (1.1), when λ is an eigenvalue of −Δ. A partial answer to this question was given in [16], where the authors showed that when n 5 and λ λ1 , then problem (1.1) has at least (n + 1)/2 pairs of nontrivial solutions. A natural question is whether all these results can be extended to the fractional nonlocal counterpart of (1.1), i.e. to the following problem

∗

(−Δ)s u = λu + |u|2 u=0

−2

u in Ω in Rn \ Ω,

(1.2)

where s ∈ (0, 1) is fixed, Ω is an open bounded subset of Rn , n > 2s, with continuous boundary, 2∗ is the fractional critical Sobolev exponent given by 2∗ := 2n/(n − 2s) and −(−Δ)s is the fractional Laplace operator which (up to normalization factors) may be defined as u(x + y) + u(x − y) − 2u(x) −(−Δ)s u(x) := dy |y|n+2s Rn

for x ∈ Rn (see [21] and references therein for further details on the fractional Laplacian). Note that, in the nonlocal setting, the condition u = 0 in Rn \Ω is the natural counterpart of the homogeneous Dirichlet boundary data u = 0 on ∂Ω, due to the nonlocal character of the problem. If we deal with the existence of nontrivial solutions for problem (1.2), a positive answer has been given in the recent papers [27,28,33], also in presence of a perturbation of the critical term (for this see, for instance, [32,34]): in all these papers the classical and well known existence results for (1.1) were extended to the nonlocal fractional setting. Other interesting existence results for nonlocal problems driven by fractional operators in a critical setting can be found in [5,24,36] and references therein. Aim of this paper is to focus the attention on the multiplicity of solutions for (1.2). In particular, our starting point will be the paper [14] and our goal will be to extend the result obtained there to the nonlocal fractional setting. Precisely, along this work we consider a generalization of (1.2), given by the following nonlocal critical problem


∗

−LK u = λu + |u|2 u=0

−2

17

u in Ω in Rn \ Ω .

(1.3)

Here LK is the nonlocal operator defined as follows: LK u(x) :=

(u(x + y) + u(x − y) − 2u(x))K(y)dy ,

x ∈ Rn ,

Rn

where the kernel K : Rn \ {0} → (0, +∞) is a function such that mK ∈ L1 (Rn ), where m(x) = min{|x|2 , 1} ;

(1.4)

there exists θ > 0 such that K(x) θ|x|−(n+2s) for any x ∈ Rn \ {0} .

(1.5)

The integrodifferential operator LK is a generalization of the fractional Laplacian, since, taking K(x) = |x|−(n+2s) , we get LK = −(−Δ)s . Of course, the trivial function u ≡ 0 is a solution of problem (1.3). Here we are interested in nontrivial solutions. Before stating our main results, we need to introduce the functional space we will work in, the variational formulation of the problem under consideration, as well as the spectrum of the operator −LK . 1.1. Notations Problem (1.3) has a variational character and the natural space where finding weak solutions for it is the functional space X0 , defined as follows (for more details we refer to [29,30], where this space was introduced and some properties of this space were proved). Along this paper the space X denotes the linear space of Lebesgue measurable functions from Rn to R such that the restriction to Ω of any function g in X belongs to L2 (Ω) and

the map (x, y) → (g(x) − g(y)) K(x − y) is in L2 (Rn × Rn ) \ (CΩ × CΩ), dxdy , where CΩ := Rn \ Ω, while X0 := {g ∈ X : g = 0 a.e. in Rn \ Ω} . We note that X and X0 are non-empty, since C02 (Ω) ⊆ X0 by [29, Lemma 5.1]: for this we need condition (1.4). The space X is endowed with the norm defined as gX := gL2 (Ω) +

Q

|g(x) − g(y)|2 K(x − y)dx dy

1/2 ,

(1.6)


18

where Q := (Rn × Rn ) \ O and O = (CΩ) × (CΩ) ⊂ Rn × Rn . Moreover, by [30, Lemma 6] as a norm on X0 we can take the function ⎛

⎞1/2

X0 g → gX0 := ⎝

|g(x) − g(y)|2 K(x − y) dx dy ⎠

.

(1.7)

Rn ×Rn

Also, (X0 , · X0 ) is a Hilbert space, as proved in [30, Lemma 7], with scalar product given by

u(x) − u(y) v(x) − v(y) K(x − y) dx dy .

X0 × X0 (u, v) → u, v X0 := Rn ×Rn

The usual fractional Sobolev space H s (Ω) is endowed with the so-called Gagliardo norm (see, for instance [1,21]) given by gH s (Ω) := gL2 (Ω) +

Ω×Ω

1/2 |g(x) − g(y)|2 dx dy . |x − y|n+2s

(1.8)

Hence, even in the model case in which K(x) = |x|−(n+2s) , the norms in (1.6) and (1.8) are not the same, because Ω × Ω is strictly contained in Q: this makes the space X0 not equivalent to the usual fractional Sobolev spaces and the classical fractional Sobolev space approach not sufficient for studying our problem from a variational point of view. The weak formulation of problem (1.3) is given by ⎧ ⎪ ⎪ (u(x) − u(y))(ϕ(x) − ϕ(y))K(x − y)dx dy − λ u(x)ϕ(x) dx ⎪ ⎪ ⎪ ⎪ ⎨Rn ×Rn Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∗

|u(x)|2

=

−2

∀ ϕ ∈ X0

u(x)ϕ(x)dx

(1.9)

Ω

u ∈ X0 ,

which represents the Euler–Lagrange equation of the functional JK, λ : X0 → R defined as 1 λ JK, λ (u) := |u(x) − u(y)|2 K(x − y) dx dy − |u(x)|2 dx 2 2 Rn ×Rn

1 − ∗ 2

Ω

∗

|u(x)|2 dx .

(1.10)

Ω

Along the present paper following problem

λk

k∈N

denotes the sequence of the eigenvalues of the


−LK u = λ u in Ω u=0 in Rn \ Ω ,

19

(1.11)

with 0 < λ1 < λ2 · · · λk λk+1 . . . λk → +∞ as k → +∞ ,

(1.12)

and with ek as eigenfunction corresponding to λk . Also, we choose ek k∈N normalized in such a way that this sequence provides an orthonormal basis of L2 (Ω) and an orthogonal basis of X0 . For a complete study of the spectrum of the integrodifferential operator −LK we refer to [27, Proposition 2.3], [31, Proposition 9 and Appendix A] and [33, Proposition 4]. Finally, we say that the eigenvalue λk , k 2, has multiplicity m ∈ N if λk−1 < λk = · · · = λk+m−1 < λk+m . In this case the set of all the eigenvalues corresponding to λk agrees with span {ek , . . . , ek+m−1 } . In the sequel we also refer to the best fractional critical Sobolev constant SK defined as follows SK :=

inf

v∈X0 \{0}

SK (v)

(1.13)

where for any v ∈ X0 \ {0} |v(x) − v(y)|2 K(x − y) dx dy SK (v) :=

Rn ×Rn

2∗

|v(x)| dx

2/2∗

.

(1.14)

Ω

Finally, in what follows, |Ω| denotes the Lebesgue measure of the set Ω. Now, we are able to state the main result of the present paper. 1.2. Main result of the paper As we said here above, the main feature of this paper concerns the existence of multiple solutions for problems (1.2) and (1.3). Precisely, with the notations introduced in Subsection 1.1, our main result reads as follows:

20


Theorem 1. Let s ∈ (0, 1), n > 2s, Ω be an open bounded subset of Rn with continuous boundary, and let K : Rn \ {0} → (0, +∞) be a function satisfying assumptions (1.4) and (1.5). Let λ ∈ R and let λ be the eigenvalue of problem (1.11) given by λ := min {λk : λ < λk } .

(1.15)

and let m ∈ N be its multiplicity. Assume that λ ∈ (λ − SK |Ω|− n , λ ) , 2s

(1.16)

where SK is the best fractional critical Sobolev constant defined in (1.13). Then, problem (1.3) admits at least m pairs of nontrivial solutions {−uλ,i , uλ,i } such that uλ,i X0 → 0

as λ → λ

for any i = 1, . . . , m. This theorem represents the fractional nonlocal counterpart of the famous multiplicity result got by Cerami, Fortunato and Struwe in [14], using essentially an abstract critical point theorem due to Bartolo, Benci and Fortunato in [6], whose main tool is a pseudo-index theory introduced in [7] for studying indefinite functionals. The weak solutions of (1.3) can be found as critical points of the Euler–Lagrange functional JK, λ associated with the equation. As usual when dealing with critical problems, one of the main difficulty in treating the problem is due to the lack of compactness that ∗ occurs. Indeed, the effect of the presence of the critical term |u|2 −2 u is that the Palais– Smale condition for JK, λ does not hold at any level, but just under a suitable threshold, which, in our case, depends on the best fractional critical Sobolev constant related to ∗ the compact embedding X0 → L2 (Ω), on s and n. Along this paper we overcome these difficulties using some strategies considered in the literature, also in contexts different from the one treated here (see, for instance, [4], where the authors studied the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by LK and involving a critical nonlinearity). In Theorem 1 we prove that, in a suitable left neighborhood of any eigenvalue of the integrodifferential operator −LK (with homogeneous Dirichlet boundary condition) the number of nontrivial solutions for problem (1.3) is, at least, twice the multiplicity of the eigenvalue. Hence, we show that there is a bifurcation from any eigenvalue of the operator −LK . In addition, we give an estimate of the length of this left neighborhood, in which the existence of multiple solutions occurs. This estimates depends on the best fractional critical Sobolev constant SK , on the Lebesgue measure of Ω, on n and s, as stated in (1.16). We would like to point out that this condition is crucial in order to show that the energy functional associated with (1.3) satisfies all the geomet-


21

ric assumptions required by the abstract critical point theorem used along the present paper. 2. A multiplicity and bifurcation result This section is devoted to the proof of Theorem 1, which is mainly based on variational and topological methods. Precisely, here we will perform the following result due to Bartolo, Benci and Fortunato (see [14, Theorem 2.5] and [6, Theorem 2.4]), which, as usual for abstract critical points theorems, gives the existence of critical points for a functional (sufficiently smooth), provided it satisfies suitable geometric and compactness conditions. Theorem 2 (Abstract critical point theorem). Let H be a real Hilbert space with norm · and suppose that I ∈ C 1 (H, R) is a functional on H satisfying the following conditions: (I1 ) I(u) = I(−u) and I(0) = 0; (I2 ) there exists a constant β > 0 such that the Palais–Smale condition for I holds in (0, β); (I3 ) there exist two closed subspaces V, W ⊂ H and positive constants ρ, δ, β , with δ < β < β, such that i) I(u) β for any u ∈ W ii) I(u) δ for any u ∈ V with u = ρ iii) codim V < +∞ and dim W codim V . Then, there exist at least dim W − codim V pairs of critical points of I, with critical values belonging to the interval [δ, β ]. The abstract result got in [6, Theorem 2.4] is a generalization of [3, Theorem 2.13], obtained using a pseudo-index theory, introduced in [7] for exploiting the existence of multiple critical points of functionals which are neither bounded above nor below on a Hilbert space. 2.1. Proof of Theorem 1 The idea consists in applying Theorem 2 to the functional JK, λ , defined in (1.10). It is easily seen that JK, λ is well defined thanks to [30, Lemma 6]. Moreover, JK, λ ∈ C 1 (X0 ) and JK, λ (u), ϕ = (u(x) − u(y))(ϕ(x) − ϕ(y))K(x − y) dx dy − λ u(x)ϕ(x) dx Rn ×Rn

−

Ω ∗

|u(x)|2 Ω

−2

u(x)ϕ(x) dx


22

for any u, ϕ ∈ X0 . Thus, critical points of JK, λ are solutions to problem (1.9), that is weak solutions for (1.3). Note also that JK, λ is even and JK, λ (0) = 0, so that condition (I1 ) of Theorem 2 is verified by JK, λ . It remains to prove that JK, λ satisfies assumptions (I2 ) and (I3 ) of Theorem 2. At this purpose, let us proceed by steps. Step 1 (Compactness of the functional JK, λ ). In the sequel we prove that, for suitable values of c, say c
0 such that |JK, λ (uj )| κ ,

(2.5)

and JK, λ (uj ),

uj κ , uj X0

(2.6)

and so 1 JK, λ (uj ) − JK, λ (uj ), uj κ (1 + uj X0 ) . 2 Furthermore, 1 JK, λ (uj ) − JK, λ (uj ), uj = 2

1 1 − ∗ 2 2

∗

uj 2L2∗ (Ω) =

∗ s uj 2L2∗ (Ω) , n

(2.7)


23

so that, thanks to (2.7), we get that for any j ∈ N ∗

uj 2L2∗ (Ω) κ∗ (1 + uj X0 )

(2.8)

for a suitable positive constant κ∗ . Consequently, recalling that 2∗ > 2 and using the Hölder inequality, we get 2/2∗

uj 2L2 (Ω) |Ω|2s/n uj 2L2∗ (Ω) κ∗

2/2∗

|Ω|2s/n (1 + uj X0 )

,

that is uj 2L2 (Ω) κ (1 + uj X0 ) ,

(2.9)

for a suitable κ > 0 not depending on j. By (2.5), (2.8) and (2.9), we have that κ JK, λ (uj ) =

∗ 1 λ 1 uj 2X0 − uj 2L2 (Ω) − ∗ uj 2L2∗ (Ω) 2 2 2

1 uj 2X0 − κ (1 + uj X0 ) , 2

with κ > 0 independent of j, and so (2.4) is proved. Now, let us show that problem (1.9) admits a solution u∞ ∈ X0 .

(2.10)

Since the sequence {uj }j∈N is bounded in X0 by (2.4), and X0 is a Hilbert space, then, up to a subsequence, still denoted by {uj }j∈N , there exists u∞ ∈ X0 such that uj → u∞ weakly in X0 , that is (uj (x) − uj (y))(ϕ(x) − ϕ(y))K(x − y) dx dy Rn ×Rn

→

(u∞ (x) − u∞ (y))(ϕ(x) − ϕ(y))K(x − y) dx dy

(2.11)

Rn ×Rn

for any ϕ ∈ X0 , as j → +∞. Moreover, by (2.4), (2.8), the embedding properties of X0 into the classical Lebesgue spaces (see [30, Lemma 8] and [32, Lemma 9]) and the fact ∗ that L2 (Rn ) is a reflexive space we have that, up to a subsequence uj → u∞ u j → u∞

∗

weakly in L2 (Rn ) 2

n

in L (R )

(2.12) (2.13)

and uj → u∞ as j → +∞.

a.e. in Rn

(2.14)


24

As a consequence of (2.4) and (2.8), we have that uj L2∗ (Ω) is bounded uniformly ∗ ∗ ∗ in j, hence the sequence {|uj |2 −2 uj }j∈N is bounded in L2 /(2 −1) (Ω), uniformly in j. Thus, (2.12) yields ∗

|uj |2

−2

∗

uj → |u∞ |2

−2

∗

weakly in L2

u∞

/(2∗ −1)

(Ω)

(2.15)

as j → +∞, and so

2∗ −2

|uj (x)|

uj (x)ϕ(x)dx →

Ω

∗

|u∞ (x)|2

−2

∗

∀ ϕ ∈ L2 (Ω)

u∞ (x)ϕ(x)dx

(2.16)

Ω

as j → +∞. Hence, in particular, we have that

2∗ −2

|uj (x)|

uj (x)ϕ(x) dx →

Ω

∗

|u∞ (x)|2

−2

u∞ (x)ϕ(x) dx ∀ ϕ ∈ X0

(2.17)

Ω ∗

as j → +∞, being X0 ⊆ L2 (Ω). By (2.3), for any ϕ ∈ X0 as j → +∞ 0 ← JK, λ (uj ), ϕ =

(uj (x) − uj (y))(ϕ(x) − ϕ(y))K(x − y) dx dy Rn ×Rn

−λ

uj (x)ϕ(x) dx −

Ω

∗

|uj (x)|2

−2

uj (x)ϕ(x) dx ,

Ω

so that, passing to the limit as j → +∞ and taking into account (2.11), (2.13) and (2.17) we get

(u∞ (x) − u∞ (y))(ϕ(x) − ϕ(y))K(x − y) dx dy − λ Rn ×Rn

−

u∞ (x)ϕ(x) dx Ω

∗

|u∞ (x)|2

−2

u∞ (x)ϕ(x) dx = 0 .

Ω

Then, u∞ is a solution of problem (1.9) and this proves (2.10). As it was proved in [27, Proposition 4.1], the function u∞ in (2.10) satisfies the following three relations,1 which will be useful in carried on our proof: ∗ s JK, λ (u∞ ) = |u∞ (x)|2 dx 0 , (2.18) n Ω

1

We will recall the proof of these relations in the Appendix A for reader’s convenience.


JK, λ (uj ) =JK, λ (u∞ ) − 1 + 2

1 2∗

25

∗

|uj (x) − u∞ (x)|2 dx Ω

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy + o(1) (2.19) Rn ×Rn

and |uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy Rn ×Rn

∗

|uj (x) − u∞ (x)|2 dx + o(1)

=

(2.20)

Ω

as j → +∞. Now, by (2.20) we get that ∗ 1 1 2 |uj (x) − u∞ (x) − uj (y) + u∞ (y)| K(x − y) dx dy − ∗ |uj (x) − u∞ (x)|2 dx 2 2 Rn ×Rn

=

=

s n

1 1 − ∗ 2 2

Ω

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy + o(1) Rn ×Rn

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy + o(1) Rn ×Rn

as j → +∞. This relation, combined with (2.19), gives JK, λ (uj ) = JK, λ (u∞ ) s + |uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy + o(1) (2.21) n Rn ×Rn

as j → +∞. Taking into account (2.2) and (2.21) we conclude that c = JK, λ (u∞ ) s + |uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy + o(1) n

(2.22)

Rn ×Rn

as j → +∞. Since the sequence {uj X0 }j∈N is bounded in R by (2.4), then we can assume that, up to a subsequence, if necessary, uj − u∞ 2X0 = |uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy → L (2.23) Rn ×Rn


26

where L ∈ [0, +∞), and so, by (2.20), we have

∗

|uj (x) − u∞ (x)|2 dx → L

(2.24)

Ω

as j → +∞. By (2.23), (2.24) and the definition of SK , we get ∗

L2/2 SK L . Then, n/(2s)

L SK

either L = 0 or

.

By (2.18), (2.22) and (2.23), we obtain c = JK, λ (u∞ ) + n/(2s)

If L SK

s s L L. n n

(2.25)

, then, by (2.25), we would get c

s n/(2s) s L SK n n

which contradicts (2.1). Thus, L = 0. As a consequence of this and of (2.23), we have that uj − u∞ X0 → 0 as j → +∞. This shows that the sequence {uj }j∈N has a (strongly) convergent subsequence. Hence, JK, λ satisfies the Palais–Smale condition at any level c, provided (2.1) is satisfied. This ends the proof of Step 1. 2 Now, let λ be as in (1.15). Then, λ = λk for some k ∈ N . Since λ has multiplicity m ∈ N by assumption, we have that λ = λ1 < λ2

if k = 1

λk−1 < λ = λk = · · · = λk+m−1 < λk+m

if k 2 .

(2.26)

Also, before going on with the proof of Theorem 1, we would note that, under condition (1.16), the parameter λ is such that λ > 0.

(2.27)


27

Indeed, by definition of λ and taking into account (1.12), it is easily seen that λ λ1 .

(2.28)

In addition, the variational characterization of the first eigenvalue λ1 (see [31, Proposition 9 and Appendix A]) gives that |u(x) − u(y)|2 K(x − y)dx dy λ1 =

min

Rn ×Rn

u∈X0 \{0}

.

(2.29)

|u(x)|2 dx Ω

Since, by Hölder inequality, it holds true that

⎞2/2∗

⎛

|u(x)|2 dx |Ω|2s/n ⎝

Ω

∗

|u(x)|2 dx⎠

,

Ω

by this and (2.29), we get λ1 SK |Ω|−2s/n , which combined with (2.28) yields λ SK |Ω|−2s/n . Hence, as a consequence of this and of (1.16), we get (2.27). Step 2 (Geometric structure of the functional JK, λ ). With the notations of the abstract result stated in Theorem 2, we set W = span {e1 , . . . , ek+m−1 } and V =

if k = 1 X0 if k 2 . u ∈ X0 : u, ej X0 = 0 ∀j = 1, . . . , k − 1

Note that both W and V are closed subset of X0 and dim W = k + m − 1

codim V = k − 1 ,

(2.30)

so that (I3 )-iii) of Theorem 2 is satisfied. In what follows, we prove that the functional JK, λ has the geometric features required by Theorem 2.


28

Proof. Let us show that the functional JK, λ verifies assumption (I3 )-i) and ii) of Theorem 2 (here condition (1.16) will be crucial). For this, let u ∈ W . Then,

u(x) =

k+m−1

ui ei (x)

i=1

with ui ∈ R, i = 1, . . . , k + m − 1. Since {e1 , . . . , ek , . . .} is an orthonormal basis of L2 (Ω) and an orthogonal one of X0 (see [31, Proposition 9]), taking into account (2.26) we get

u2X0 =

k+m−1

k+m−1

u2i ei 2X0 =

i=1

λi u2i λk

k+m−1

i=1

u2i = λk u2L2 (Ω) = λ u2L2 (Ω) ,

i=1

so that, by this and Hölder inequality, we have JK, λ (u) =

1 2 −

|u(x) − u(y)|2 K(x − y) dx dy − Rn ×Rn

1 2∗

|u(x)|2 dx Ω

∗

|u(x)|2 dx Ω

λ 2

1 (λ − λ) 2

|u(x)|2 dx − Ω

1 2∗

Ω

⎛

2s 1 (λ − λ)|Ω| n ⎝ 2

∗

|u(x)|2 dx

∗

⎞ 22∗

|u(x)| dx⎠ 2

1 − ∗ 2

Ω

Ω

Now, for t 0 let g(t) =

2s 1 1 ∗ (λ − λ)|Ω| n t2 − ∗ t2 . 2 2

Note that the function g is differentiable in (0, +∞) and ∗

g (t) = (λ − λ)|Ω| n t − t2 2s

−1

so that g (t) 0 if and only if ∗ 2s 1/(2 −2) t t¯ = (λ − λ)|Ω| n .

∗

|u(x)|2 dx.

(2.31)


29

As a consequence of this, t¯ is a maximum point for g and so for any t 0 n s g(t) max g(t) = g(t¯) = (λ − λ) 2s |Ω| . t0 n

(2.32)

By (2.31) and (2.32) we get sup JK, λ (u) max g(t) = t0

u∈W

n s (λ − λ) 2s |Ω| . n

(2.33)

We observe that n s (λ − λ) 2s |Ω| > 0 n

since λ < λ by (1.16). Finally, let u ∈ V . Then, u2X0 λ u2L2 (Ω)

(2.34)

Indeed, if u ≡ 0, then the assertion is trivial, while if u ∈ V \ {0} it follows from the variational characterization of λ = λk given by |u(x) − u(y)|2 K(x − y)dx dy λk =

Rn ×Rn

min

u∈V \{0}

, |u(x)|2 dx

Ω

as proved in [31, Proposition 9]. Thus, by (1.13), (2.34) and taking into account that λ > 0 (see (2.27)), it follows that ∗ λ 1 1 − u2X0 − u2X0 1/2 ∗ λ 2 SK 1 λ 1 2∗ −2 2 1− − = uX0 uX0 . 1/2 2 λ 2∗ S

JK, λ (u)

1 2

(2.35)

K

Now, let u ∈ V be such that uX0 = ρ > 0. Since 2∗ > 2, we can choose ρ sufficiently small, say ρ ρ¯ with ρ¯ > 0, so that 1 2

1−

λ λ

−

1 1/2 2 ∗ SK

∗

ρ2

−2

>0

(2.36)


30

and n λ 1 λ s 1 ρ2 2∗ −2 1− − 1 − < (λ − λ) 2s |Ω| . ρ ρ < 1/2 2 λ 2 λ n 2 ∗ SK 2

2

(2.37)

Now, we can conclude the proof of Theorem 1. By Step 1 it easily follows that JK, λ satisfies (I2 ) with β=

s n/(2s) S > 0. n K

Also, by Step 2 (see (2.33)–(2.37)) we get that JK, λ verifies (I3 ) with ρ = ρ¯ , β =

n s (λ − λ) 2s |Ω| n

and 1 λ 1 2∗ −2 1− − δ = ρ¯ ρ¯ . 1/2 2 λ 2 ∗ SK 2

Note that 0 < δ < β < β thanks to (2.36), (2.37) and assumption (1.16). All in all, the functional JK, λ satisfies both the compactness assumption and the geometric features required by the abstract critical points theorem stated in Theorem 2. As a consequence, JK, λ has m pairs {−uλ,i , uλ,i } of critical points whose critical value JK, λ (±uλ,i ) is such that n 1 λ 1 s 2∗ −2 1− − 0 < ρ¯ ρ¯ JK, λ (±uλ,i ) (λ − λ) 2s |Ω| 1/2 X0 ∗ 2 λ n 2 SK 2

(2.38)

for any i = 1, . . . , m. Since JK, λ (0) = 0 and (2.38) holds true, it is easy to see that these critical points are all different from the trivial function. Hence, problem (1.3) admits m pairs of nontrivial weak solutions. Now, fix i ∈ {1, . . . , m}. By (2.38) we obtain (λ − λ)n/2s |Ω|

s JK, λ (uλ,i ) n = JK, λ (uλ,i ) −

1 J (uλ,i ), uλ,i 2 λ


= =

1 1 − 2 2∗

31

∗

uλ,i 2L2∗ (Ω)

∗ s uλ,i 2L2∗ (Ω) , n

(2.39)

so that, passing to the limit as λ → λ in (2.39), it follows that ∗

uλ,i 2L2∗ (Ω) → 0 as λ → λ .

(2.40)

∗

Then, by (2.40), since L2 (Ω) → L2 (Ω) continuously (being Ω bounded), we also get uλ,i 2L2 (Ω) → 0 as λ → λ .

(2.41)

So, arguing as above, we have (λ − λ)n/2s |Ω|

∗ s 1 λ 1 JK, λ (uλ,i ) = uλ,i 2X0 − uλ,i 2L2 (Ω) − ∗ uλ,i 2L2∗ (Ω) , n 2 2 2

which combined with (2.40) and (2.41) gives uλ,i X0 → 0

as λ → λ .

This concludes the proof of Theorem 1. Conflict of interest statement The authors declare that they have no conflict of interest. Acknowledgements We would like to thank the anonymous referees for their useful comments. Appendix A Here we give the details of the proof of relations (2.18)–(2.20), which were proved in [27, Proposition 4.1], in order to make the present paper self-contained and for reader’s convenience. Proof of (2.18). By (2.10), taking ϕ = u∞ ∈ X0 as a test function in (1.9), we get

|u∞ (x) − u∞ (y)|2 K(x − y)dx dy − λ Rn ×Rn

|u∞ (x)|2 dx =

Ω

∗

|u∞ (x)|2 dx , Ω


32

so that JK, λ (u∞ ) =

1 1 − ∗ 2 2

∗

|u∞ (x)|2 dx =

s n

Ω

This concludes the proof of (2.18).

∗

|u∞ (x)|2 dx 0 . Ω

2

Proof of (2.19). At this purpose, by (2.4) and taking into account the embedding properties of X0 into the Lebesgue spaces (see [30, Lemma 8] and [32, Lemma 9]), the sequence ∗ {uj }j∈N is bounded in X0 and in L2 (Ω). Thus, since (2.14) holds true, by the Brezis–Lieb Lemma (see [9, Theorem 1]), we get |uj (x) − uj (y)|2 K(x − y) dx dy Rn ×Rn

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy

= Rn ×Rn

|u∞ (x) − u∞ (y)|2 K(x − y) dx dy + o(1)

+

(A.1)

Rn ×Rn

and

∗

|uj (x)|2 dx = Ω

∗

|uj (x) − u∞ (x)|2 dx + Ω

∗

|u∞ (x)|2 dx + o(1)

(A.2)

Ω

as j → +∞. Now, by (2.13), (A.1) and (A.2) we deduce that 1 JK, λ (uj ) = 2

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy Rn ×Rn

1 + 2

λ |u∞ (x) − u∞ (y)| K(x − y) dx dy − 2

Rn ×Rn

1 − ∗ 2

∗

|uj (x) − u∞ (x)|2 Ω

1 = JK, λ (u∞ ) + 2 −

|u∞ (x)|2 dx

2

1 2∗

1 dx − ∗ 2

Ω ∗

|u∞ (x)|2 dx + o(1) Ω

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y) dx dy Rn ×Rn ∗

|uj (x) − u∞ (x)|2 dx + o(1) Ω

as j → +∞, which gives (2.19).

2


33

Proof of (2.20). For this, note that, as a consequence of (2.12), (2.15) and (A.2), we get

∗

|uj (x)|2

Ω

−2

∗

uj (x) −|u∞ (x)|2

∗

|uj (x)|2 dx −

= Ω

∗

|uj (x)|2

−2

−2

Ω

Ω

∗

|u∞ (x)|2 dx Ω

∗

uj (x) − u∞ (x) dx

u∞ (x)uj (x) dx

uj (x)u∞ (x) dx +

|uj (x)|2 dx −

=

∗

|u∞ (x)|2

u∞ (x)

Ω

−

−2

∗

|u∞ (x)|2 dx + o(1) Ω

∗

|uj (x) − u∞ (x)|2 dx + o(1)

=

(A.3)

Ω

as j → +∞. By (2.3), (2.4) and (2.10), we have that o(1) = JK, λ (uj ), uj − u∞ = JK, λ (uj ) − JK, λ (u∞ ), uj − u∞

(A.4)

as j → +∞. On the other hand, by (2.13) and (A.3), it holds true that JK, λ (uj ) − JK, λ (u∞ ), uj − u∞ = |uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y)dx dy Rn ×Rn

−λ

|uj (x) − u∞ (x)|2 dx Ω

−

∗

|uj (x)|2

−2

∗

uj (x) − |u∞ (x)|2

−2

u∞ (x) (uj (x) − u∞ (x)) dx

Ω

|uj (x) − u∞ (x) − uj (y) + u∞ (y)|2 K(x − y)dx dy

= Rn ×Rn

−

∗

|uj (x) − u∞ (x)|2 dx + o(1)

(A.5)

Ω

as j → +∞. Then, combining (A.4) and (A.5), we get (2.20).

2

34


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