Multiple Solutions of Neumann Problems: An Orlicz ...

Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting

Ghasem A. Afrouzi, Vicenţiu D. Rădulescu & Saeid Shokooh

Bulletin of the Malaysian Mathematical Sciences Society ISSN 0126-6705 Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-015-0153-x

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Author's personal copy Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-015-0153-x

Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting Ghasem A. Afrouzi1 · Vicen¸tiu D. R˘adulescu2,3 · Saeid Shokooh1,4

Received: 25 February 2015 / Revised: 17 June 2015 © Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Abstract In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri. Keywords Three solutions · Non-homogeneous differential operator · Orlicz–Sobolev space Mathematics Subject Classification 46N20 · 58E05

Primary 34B37 · Secondary 35J60 · 35J70 ·

Communicated by Norhashidah Hj. Mohd. Ali.

B

Vicen¸tiu D. R˘adulescu [email protected]; [email protected] Ghasem A. Afrouzi [email protected] Saeid Shokooh [email protected]

1

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

2

Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania

3

Topology, Geometry and Nonlinear Analysis Group, Department of Mathematics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia

4

Department of Mathematics, Faculty of Sciences, University of Gonbad Kavous, Gonbad Kavous, Iran

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Author's personal copy G. A. Afrouzi et al.

1 Introduction In this paper, we are concerned with the qualitative analysis of weak solutions for a large class of non-linear elliptic equations with Neumann boundary condition. The main features of this paper are the following: (i) the presence of a non-homogeneous differential operator and the treatment in a suitable Orlicz–Sobolev function space; (ii) the use of the Ricceri three-critical point theorem, which is a powerful analytic tool for multiplicity results in non-linear problems with a variational structure. Orlicz–Sobolev spaces have been used in the last decades to model various phenomena. These function spaces play a significant role in many fields of mathematics, such as approximation theory, partial differential equations, calculus of variations, non-linear potential theory, the theory of quasi-conformal mappings, non-Newtonian fluids, image processing, differential geometry, geometric function theory, and probability theory. These spaces consist of functions that have weak derivatives and satisfy certain integrability conditions. We refer to Chen, Levine and Rao [1], who proposed a framework for image restoration based on a Laplace operator with variable exponent. A second major application of non-homogeneous differential operators with variable exponent is the modeling of some materials with inhomogeneities, for instance, electrorheological (non-Newtonian) fluids (sometimes referred to as ‘smart fluids’), cf. [2–7]. Materials requiring such more advanced theory have been studied experimentally since the middle of the last century. The first major discovery in electrorheological fluids is due to Willis Winslow in 1949. These fluids have the interesting property that their viscosity depends on the electric field in the fluid. Winslow noticed that in such fluids (for instance, lithium polymethacrylate) viscosity in an electrical field is inversely proportional to the strength of the field. The field induces string-like formations in the fluid, which are parallel to the field. They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known as the Winslow effect. For a general account of the underlying physics we refer to Halsey [8] and Pfeiffer et al. [9]. An overview of Orlicz–Sobolev spaces is given in the monographs by Rao and Ren [10] and R˘adulescu and Repovš [6]. On the other hand, the study of differential equations and variational problems with non-linear boundary condition have been studied recently in many papers, see [11–16]. The classical three-critical point theorem of Pucci and Serrin [17,18] asserts that if f : X → R is of class C 1 (X is a Banach space), satisfies the Palais–Smale condition, and has two local minima, then f has a third critical point. The general variational principle of Ricceri [19,20] extends the Pucci–Serrin theorem and provides alternatives for the multiplicity of critical points of certain functionals depending on a parameter. We refer to [21] for several applications of the Ricceri variational principles. In this work, we are concerned with the following problem involving nonhomogeneous differential operators: −div α(x, |∇u(x)|)∇u(x) + α x, |u(x)| u(x) = λ f x, u(x) for x ∈ , f,g ∂u Nλ,μ α x, |∇u(x)| for x ∈ ∂, (x) = μg γ u(x) ∂ν

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Author's personal copy Multiple Solutions of Neumann Problems: An Orlicz–Sobolev…

where is a bounded domain in R N (N ≥ 3) with smooth boundary ∂, ∂u ∂ν is the outer unit normal derivative, f : × R → R is a Carathéodory function, g : R → R is a continuous function, λ is a positive parameter, μ is a non-negative parameter, and ¯ × R → R and γ will be specified later. the functions α(x, t) : In this paper, motivated by the above facts and the recent papers [22–26], we f,g establish some new sufficient conditions under which the problem (Nλ,μ ) possesses three weak solutions in the Orlicz–Sobolev space. At first, we present two three solutions existence results under algebraic conditions on f (see Theorems 3.1, 3.2). Next, assuming that the growth of f is sublinear at infinity, we establish the existence of f,g three weak solutions for the problem (Nλ,μ ) (see Theorem 3.3). This paper is organized as follows. In Sect. 2, some preliminaries and the abstract Orlicz–Sobolev spaces setting are presented. In Sect. 3, we discuss the existence of f,g three weak solutions for the problem (Nλ,μ ). We also point out special cases of the results, and we illustrate the results by presenting an example.

2 Functional Setting We now state two critical point theorems which are the main tools for the proofs of our results. The first results of this type have been obtained in [19]. In the first result, the coercivity of the functional J − λI is required; in the second one, a suitable sign hypothesis is assumed. Theorem 2.1 is a particular case of Theorem 1 of Ricceri [20], while for Theorem 2.2, we refer to Bonanno and Candito [27, Corollary 3.1]. Theorem 2.1 Let X be a reflexive real Banach space; J : X → R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X ∗ , I : X → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, such that J (0) = I (0) = 0 . Assume that there exist r > 0 and x¯ ∈ X , with r < J (x) ¯ such that (i) sup J (x)≤r I (x) < r I (x)/J ¯ (x), ¯ (ii) for each λ in r :=

J (x) ¯

,

r

I (x) ¯ sup J (x)≤r I (x)

,

the functional J − λI is coercive. Then, for each λ ∈ r the functional J − λI has at least three distinct critical points in X . Theorem 2.2 Let X be a reflexive real Banach space; J : X → R be a convex, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X ∗ , I : X → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, such that

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inf J = J (0) = I (0) = 0 . X

Assume that there exist two positive constants r1 , r2 > 0 and x¯ ∈ X , with 2r1 < J (x) ¯ < r22 , such that (j)

sup J (x)≤r1 I (x) r1 sup J (x)≤r2 I (x) r2

0 whenever t > 0, limt→∞ (x, t) = ∞, ( 2 ) for every t ≥ 0, (·, t) : → R is a measurable function. Since ϕ(x, ·) satisfies condition (ϕ), we deduce that (x, ·) is convex and increasing from R+ to R+ .

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For the function , we define the generalized Orlicz class,

K () = u : → R, measurable;

(x, |u(x)|) dx < ∞

and the generalized Orlicz space,

L () = u : → R, measurable; lim

(x, λ|u(x)|) dx = 0 . λ→0+

The space L () is a Banach space endowed with the Luxemburg norm

|u(x)| dx ≤ 1 |u| = inf μ > 0;

x, μ or the equivalent norm (the Orlicz norm)

|u|( ) = sup uv dx ; v ∈ L (),

(x, |v(x)|) dx ≤ 1 ,

where denotes the conjugate Young function of , that is,

(x, t) = sup ts − (x, s); s ∈ R ,

∀ x ∈ , t ≥ 0 .

s>0

Furthermore, for and conjugate Young functions, the Hölder type inequality holds true uv dx ≤ B × |u| × |v| , ∀ u ∈ L (), v ∈ L () , (2.1)

where B is a positive constant (see [30], Theorem 13.13). In this paper we assume that there exist two positive constants ϕ0 and ϕ 0 such that 1 < ϕ0 ≤

tϕ(x, t) ≤ ϕ 0 < ∞,

(x, t)

∀ x ∈ , t ≥ 0 .

(2.2)

The above relation implies that satisfies the 2 -condition, i.e.

(x, 2t) ≤ K × (x, t),

∀ x ∈ , t ≥ 0 ,

(2.3)

where K is a positive constant (see [31, Proposition 2.3]). Relation (2.3) and Theorem 8.13 in [30] imply that L () = K (). Furthermore, we assume that satisfies the following condition: for each x ∈ , the function [0, ∞) t → (x,

√

t) is convex .

(2.4)

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Relation (2.4) assures that L () is an uniformly convex space and thus, a reflexive space (see [31, Proposition 2.2]). On the other hand, we point out that assuming that and belong to class and (x, t) ≤ K 1 × (x, K 2 × t) + η(x),

∀ x ∈ , t ≥ 0 ,

(2.5)

where η ∈ L 1 (), η(x) ≥ 0 a.e. x ∈ and K 1 , K 2 are positive constants, then by Theorem 8.5 in [30] there exists the continuous embedding L () ⊂ L (). Next, we define the generalized Orlicz–Sobolev space

∂u W 1, () = u ∈ L (); ∈ L (), ∂ xi

i = 1, . . . , N .

On W 1, () we define the equivalent norms u1, = | |∇u| | + |u| , u2, = max | |∇u| | , |u| ,

|u(x)| |∇u(x)|

x, + x, dx ≤ 1 . u = inf μ > 0; μ μ More precisely, for every u ∈ W 1, (), we have u ≤ 2u2, ≤ 2u1, ≤ 4u

(2.6)

(see [31, Proposition2.4]). The generalized Orlicz–Sobolev space W 1, () endowed with one of the above norms is a reflexive Banach space. In the following, we will use the norm · on E := W 1, () and we suppose that γ : E → L () is the trace operator. The following lemma is useful in the proof of our results. Lemma 2.3 Let u ∈ E. Then

x, |∇u(x)| + x, |u(x)| dx ≥ uϕ0

0

x, |∇u(x)| + x, |u(x)| dx ≥ uϕ

if u > 1;

(2.7)

if u < 1.

(2.8)

For the proof of the previous result see, for instance, Lemma 2.3 of [32]. We point out that assuming that and belong to class , satisfying relation (2.5) and inf x∈ (x, 1) > 0, inf x∈ (x, 1) > 0 then there exists the continuous embedding W 1, () → W 1, (). f,g In this paper we study the problem (Nλ,μ ) in the particular case when satisfies M × |t| p(x) ≤ (x, t),

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∀ x ∈ , t ≥ 0 ,

(2.9)


where p(x) ∈ C() with p − := inf x∈ p(x) > N for all x ∈ , and M > 0 is a constant. By the relation (2.9) we deduce that E is continuously embedded in W 1, p(x) () (see relation (2.5) with (x, t) = |t| p(x) ). Moreover, as pointed out in [33] and [34], W 1, p(x) () is continuously embedded in − − 1, W p () and since p − > N , we deduce that W 1, p () is compactly embedded in 0 0 C (). Thus, E is compactly embedded in C (), and there exists a constant m > 0 such that (2.10) u∞ ≤ m u, ∀ u ∈ E, where u∞ := supx∈ |u(x)|. Example 2.4 Define ϕ(x, t) = p(x)

|t| p(x)−2 t log 1 + |t|

for t = 0, and ϕ(x, 0) = 0,

where p(x) ∈ C() satisfying N < p(x) < +∞ for all x ∈ . Some simple computations imply |t| p(x) +

(x, t) = log 1 + |t|

0

|t|

s p(x) 2 ds, (1 + s)(log 1 + s)

and the relations (ϕ), ( 1 ), and ( 2 ) are verified. For each x ∈ fixed, by Example 3 on p. 243 in [14], we have p(x) − 1 ≤

tϕ(x, t) ≤ p(x),

(x, t)

∀ t ≥ 0.

Thus, the relation (2.2) holds true with ϕ0 = p − − 1 and ϕ 0 = p + := supx∈ p(x). Next, satisfies the condition (2.9) since

(x, t) ≥ t p(x)−1 ,

∀ x ∈ , t ≥ 0 .

Finally, we point out that trivial computations imply that x ∈ and t ≥ 0. Thus, the relation (2.4) is satisfied.

√ d 2 (x, t ) dt 2

≥ 0 for all

f,g

We say that u ∈ E is a weak solution of the problem (Nλ,μ ) if

α x, |∇u(x)| ∇u(x) × ∇v(x) dx + α x, |u(x)| u(x)v(x) dx

=λ f x, u(x) v(x) dx + μ g γ u(x) γ v(x) dσ

∂

for every v ∈ E.

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3 Main Results recall that f : × R → R is a Carathéodory function and F(x, t) := We t f 0 (x, s) ds. Fix d = 0 and c ≥ m [see relation (2.10)] such that F(x, d) dx . <

(x, |d|) dx

max|t|≤c F(x, t) dx c ϕ0 m

Pick λ ∈ 1 :=

(x, |d|) dx

F(x, d) dx

c ϕ0

,

m

max|t|≤c

,

F(x, t) dx

(3.1)

and set cϕ0 − λm ϕ0 max|t|≤c F(x, t) dx (x, |d|) dx − λ F(x, d) dx δ1 := min , m ϕ0 a(∂) max|t|≤c G(t) a(∂)G(d)

(3.2) and

⎧ ⎪ ⎪ ⎨

1

δ 1 := min δ1 , ⎪ ⎪ ⎩ max 0, 2m ϕ0 a(∂) lim sup|ξ |→+∞

G(ξ ) |ξ |ϕ0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

(3.3)

where a(∂) := ∂ dσ , and we read 01 = +∞ whenever this case occurs. With the above notations we establish the following result. Theorem 3.1 Suppose that there exist d = 0 and c ≥ m, with

c ϕ0

(x, |d|) dx >

m

,

such that (A1)

max|t|≤c F(x, t) dx c ϕ0 m

F(x, d) dx ; < (x, |d|) dx

(A2) supx∈ F(x, ξ ) lim sup < |ξ |ϕ0 |ξ |→+∞

max|t|≤c F(x, t) dx . 2cϕ0 ||

Then, for every λ ∈ 1 , where 1 is given by (3.1), and for every continuous function g : R → R such that

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lim sup

|ξ |→+∞

G(ξ ) < +∞, |ξ |ϕ0 f,g

there exists δ 1 > 0 given by (3.3) such that, for each μ ∈ [0, δ 1 [, the problem (Nλ,μ ) admits at least three weak solutions in E. Proof Fix λ, μ and g as in the conclusion. Our aim is to apply Theorem 2.1. For each u ∈ E, let the functionals J, I : E → R be defined by

(x, |∇u(x)|) + (x, |u(x)|) dx, J (u) :=

μ I (u) := F(x, u(x)) dx + G(γ (u(x))) dσ, λ ∂ and put Tλ,μ (u) := J (u) − λI (u). Similar arguments as those used in [31, Lemma 4.2] imply that J ∈ C 1 (E, R) with the derivative given by

α x, |∇u(x)| ∇u(x) × ∇v(x) dx + α x, |u(x)| u(x) v(x) dx J (u), v =

for every v ∈ E. Also J is bounded from below. Moreover, I ∈ C 1 (E, R) and

μ f x, u(x) v(x) dx + g γ u(x) γ v(x) dσ I (u), v = λ ∂ for every v ∈ E. So, with standard arguments, we deduce that the critical points of the functional f,g Tλ,μ are the weak solutions of problem (Nλ,μ ). We will verify (i) and (ii) of Theorem 2.1. Let w be the function defined by w(x) := d for all x ∈ and put r :=

c ϕ0 m

.

Clearly, w ∈ E and from the condition

J (w) =

c ϕ0

x, |d| dx > , one has m

x, |d| dx > r.

Also, we have

I (w) =

F(x, d) dx +

μ a(∂)G(d). λ

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0 By Lemma 2.3 and the fact max r 1/ϕ0 , r 1/ϕ = r 1/ϕ0 , we deduce

c . u ∈ E : J (u) < r ⊆ u ∈ E : u < r 1/ϕ0 = u ∈ E : u < m

Moreover, due to (2.10), we have |u(x)| ≤ u∞ ≤ mu ≤ c,

∀x ∈ .

Hence,

c ⊆ u ∈ E : u∞ ≤ c . m

u ∈ E : u
, J (w) λ

(3.5)


while, if G(d) < 0, it holds since μ
0. |ξ |ϕ0 |ξ |→+∞ lim sup

So, we can fix ε > 0 satisfying supx∈ F(x, ξ ) lim sup 1, we obtain Tλ,μ (u) ≥

1 uϕ0 − λθ1 (u + us+1 ) − μh μ a(∂) 2

for any u ∈ E with u > 1. Since 1 < s + 1 < ϕ0 it follows that lim

u→+∞

Tλ,μ (u) = +∞

for every λ > 0 and μ ≥ 0. Hence, Tλ,μ is a coercive functional. Then, also condition (ii) holds. Since all the assumptions of Theorem 2.1 are satisfied, for each λ ∈ 3 is given by (3.10) there exists δ 3 given by (3.12) such that, for each μ ∈ [0, δ 3 [ the functional Tλ,μ has at least three distinct critical points in E, which are the weak f,g solutions of the problem (Nλ,μ ). The proof is complete. A particular case of Theorem (3.3) is established in the following result. Theorem 3.4 Let b : → R be a bounded measurable and positive function and ξ f : R → R be a continuous and non-negative function. Set F(ξ ) := 0 f (t) dt. Further, suppose that there exist d = 0 and c < m, with

(x, |d |) dx >

c ϕ 0 m

,

such that (D1) F(c ) F(d ) ; < ϕ 0 c x, |d | dx m

(D2) There exist c0 > 0 and 0 < s < ϕ0 − 1 such that | f (t)| ≤ c0 (1 + |t|s ) for every t ∈ R. Then, for every λ belonging to ⎤

(x, |d |) dx

ϕ 0

⎡

c m

⎢ ⎥ ⎢ ⎥ ⎦ b 1 F(d ) , b 1 F(c ) ⎣ , L () L () and for every continuous function g : R → R such that lim sup

|ξ |→+∞

G(ξ ) < +∞, |ξ |ϕ0

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there exists δ 4 > 0 defined by ⎧ ⎪ ⎪ ⎨ 1

δ 4 := min δ4 , ⎪ ⎪ ⎩ max 0, 2m ϕ0 a(∂) lim sup|ξ |→+∞

G(ξ ) |ξ |ϕ0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

where ⎧ ⎫ ⎨ c ϕ 0 − λm ϕ 0 b 1 F(c ) x, |d | dx − λb 1 F(d ) ⎬ L () L () δ4 := min , , ⎭ ⎩ m ϕ 0 a(∂) max|t|≤c G(t) a(∂)G(d ) for each μ ∈ [0, δ 4 [, the problem

−div α(x, |∇u(x)|)∇u(x) + α x, |u(x)| u(x) = λb(x) f u(x) for x ∈ , ∂u for x ∈ ∂ α(x, |∇u(x)|) (x) = μg γ u(x) ∂ν

b f,g Nλ,μ

possesses at least three weak solutions in E. Remark 3.5 The same conclusion of Theorem 3.4 holds under the assumption that b : → R is a bounded measurable function with ess inf x∈ b(x) ≥ 0 and b(x) dx > 0. A direct consequence of the previous result reads as follows. Corollary 3.6 Let b : → R be a bounded measurable and positive function. Moreover, let f : R → R be a non-negative (not identically zero) and continuous function such that f (t) lim 0 = 0. (l0 ) t→0+ t ϕ −1 Further, assume that condition (C2) holds. Then, for each

(x, |ρ|) dx 1 λ> , inf b L 1 () ρ∈S F(ρ) where S := {ρ > 0 : F(ρ) > 0}, the following problem −div α(x, |∇u(x)|)∇u(x) + α(x, |u(x)|)u(x) = λb(x) f (u(x)) for x ∈ , bf ∂u Nλ (x) = 0 for x ∈ ∂ ∂ν possesses at least three weak solutions in E. Proof Fix λ>

123

1 b L 1 ()

inf

ρ∈S

(x, |ρ|) dx

F(ρ)

.


Then there exists ρ such that F(ρ) > 0 and λ>

1 b L 1 ()

(x, |ρ|) dx

F(ρ)

.

By using condition (l0 ) one has lim

F(ξ )

ξ →0+

|ξ |ϕ

0

= 0.

Therefore, we can find a positive constant c such that ⎧ ⎫ 1/ϕ 0 ⎬ ⎨ c < m min 1,

(x, |ρ|) dx , ⎩ ⎭ and F(c) cϕ

0

1, t where s ∈]0, ϕ0 − 1[. Further, let b : → R be a bounded measurable and positive function. From Corollary 3.6, for each parameter λ>

1 b L 1 ()

inf

ρ>0

(x, |ρ|) dx

F(ρ)

,

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( ) ⎧ p(x)−2 ∇u ⎪ |∇u| |u| p(x)−2 u ⎪ ⎨ −div p(x) = λb(x) f (u(x)) + p(x) log(1 + |∇u|) log(1 + |u|) ⎪ ⎪ ∂u ⎩ (x) = 0 ∂ν

for x ∈ , for x ∈ ∂

admits at least three non-negative weak solutions. Acknowledgments V. R˘adulescu acknowledges the support through Grant Advanced Collaborative Research Projects CNCS-PCCA-23/2014.

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