GENERAL QUASILINEAR PROBLEMS INVOLVING p(x)−LAPLACIAN WITH ROBIN BOUNDARY CONDITION

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving p(x)-Laplace type equation, namely { −div (a(|∇u|p(x))|∇u|p(x)−2∇u) = λf(x, u) in Ω, n · a(|∇u|p(x))|∇u|p(x)−2∇u+ b(x)|u|p(x)−2u = g(x, u) on ∂Ω. Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.


Introduction
In this paper we study the nonlinear elliptic boundary value problem with Robin conditions −div (a(|∇u| p(x) )|∇u| p(x)−2 ∇u) = λf (x, u) in Ω, n · a(|∇u| p(x) )|∇u| p(x)−2 ∇u + b(x)|u| p(x)−2 u = g(x, u) on ∂Ω, (1.1) where Ω is an open bounded subset of R N (N ≥ 2), with smooth boundary, n is the outer unit normal vector on ∂Ω, b is a positive continuous function defined on R N , p ∈ C + (Ω) with 1 < p − := inf Ω p(x) ≤ p + := sup Ω p(x) < N and p(x) < p * (x) where for any x ∈ Ω. It is clear that the equation in question is elliptic since it describes phenomena that do not change from moment to moment, and that the operator Lu = −div (a(|∇u| p(x) )|∇u| p(x)−2 ∇u) is an elliptic operator in divergence form.
Recently, the study of differential equations and variational problems involving p(x)-growth conditions have been extensively investigated and received much attention because they can be presented as models for many physical phenomena which arouse in the study of elastic mechanics [32], electro-rheological fluid dynamics [27] and image processing [6], electrical resistivity and polycrystal plasticity [3,4] and continuum mechanics [2] etc, for an overview of this subject, and for more details we refer readers to [11] and [5,10] and the references therein. The existence of nontrivial solutions to nonlinear elliptic boundary value problems has been extensively studied by many researchers [1,7,14,15,18,21,23,24] and references therein.
It is known that the extension p(x)-Laplace operator possesses more complicated structure than the p-Laplacian. For example, it is inhomogeneous and usually it does not have the so-called first eigenvalue, since the infimum of its spectrum is zero.
However, to understand the role of the variable exponent, well, although most of the materials can be accurately modeled with the help of the classical Lebesgue and Sobolev spaces L p and W 1,p , where p is a fixed constant, there are some nonhomogeneous materials, for which this is not adequate, e.g. the rheological fluids mentioned above, which are characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field. Thus it is necessary for the exponent p to be nonstandard, therefore, the spaces with variable exponents are required. As an introduction and a history coverage to the subject of variable exponent problems, we advice the reader to see the monograph [12] and the articles [16,20].
Note that, the p(x)-Laplace operator in (1.1) is a special case of the divergence form operator −div (a(|∇u| p(x) )|∇u| p(x)−2 ∇u) which appears in many nonlinear diffusion problems, in particular in the mathematical modeling of non-Newtonian fluids. When we have the generalized Capillary operator (which is essential in applied fields like industrial, biomedical and pharmaceutical) initiated by W. Ni and J. Serrin [22]. Inspired by the works in [25] and [19], we study the existence and multiplicity of nontrivial solutions the problem (1.1), via the mountain pass theorem and the Rabinowitz's symmetric mountain pass theorem [26].
We assume the following conditions: The function a : R + → R is continuous and the mapping Θ : R N → R, given by Θ(ξ) = A(|ξ| p(x) ) is strictly convex, where A is the primitive of a, that is (A 1 ) There exist two constants 0 < L < K such that L ≤ a(t) ≤ K for all t ≥ 0.
We assume that f, g : Ω × R → R are of Carathéodory functions, f (x, ·) = g(x, ·) = 0 and satisfy: where f 1 is nonnegative, measurable function and where f 2 > 0 in some nonempty open set O ⊂ Ω; and there exists a positive constants C g such that 0 ≤ g 1 ≤ C g ; The main result of this paper is as follow.
Then there exists λ * > 0 such that for every λ ∈]0, λ * [, the problem (1.1) admits at least one nontrivial solution. In addition, if we assume the following conditions: (G 4 ) the functions f and g are even, then the problem (1.1) has infinitely many solutions for every λ > 0.
The remainder of this paper is organized as follows, in Section 2 we introduce some technical results and required hypotheses in order to solve our problem, in Section 3 we state some and prove the main results of this work.

Preliminaries
In the sequel, let p(x) ∈ C + (Ω), where The variable exponent Lebesgue space is defined by Remark 1. Variable exponent Lebesgue spaces resemble to classical Lebesgue spaces in many respects, they are separable Banach spaces and the Hölder inequality holds. The inclusions between Lebesgue spaces are also naturally generalized, that is, if 0 < mes (Ω) < ∞ and p, q are variable exponents such that p(x) < q(x) a. e. in Ω, then there exists a continuous embedding The variable exponent Sobolev space is defined by Proposition 1 [16,17]. The spaces L p(x) (Ω) and W 1,p(x) (Ω) are separable, uniformly convex, reflexive Banach spaces. The conjugate space of (Ω), the following inequality holds (see [15,Proposition 2.5]) Proposition 2 [13]. Let p(x) and q(x) be measurable functions such that p(x) ∈ L ∞ (Ω) and In particular if p(x) = p is a constant, then Proposition 3 [16,17]. Assume that the boundary of Ω possesses the cone property and Proposition 4 [9]. For p ∈ C + (Ω) and such r ∈ C + (∂Ω) that r(x) ≤ p ∂ (x) (r(x) < p ∂ (x)) for all x ∈ Ω, there is a continuous (compact) embedding Then u ∂ is a norm on W 1,p(x) (Ω) which is equivalent to and dσ x is the measure on the boundary ∂Ω. Then by Proposition 5, · is also a norm on W 1,p(x) (Ω) which is equivalent to · W 1,p(x) (Ω) and · ∂ , the proof of this statement can be found in [8, p. 551]. Now, we introduce the modular ρ : X → R defined by Here, we give some relations between the norm || · || and the modular ρ.
(Ω) and maps bounded set into bounded set.
As a consequence of Proposition 7, the Carathéodory function f defines an operator N f which is called the Nemytskii operator.
Now we introduce the Euler-Lagrange functional I λ : X −→ R associated with problem (1.1) defined by Furthermore, the (weak) solutions of (1.1) are precisely the critical points of the functional I λ .
Lemma 1 [31] . Let Then the mapping L : X → X * is a strictly monotone, continuous bounded homeomorphism and is of type (S + ), namely assumptions u n ⇀ u and lim sup n→+∞ L(u n )(u n − u) ≤ 0, imply u n → u.
By Proposition 7, we can see that the functional I λ is well defined on X and I λ ∈ C 1 (X, R) with its Fréchet derivative is giving by Let X be a real Banach space and let be a functional I ∈ C 1 (X, R). We say that I satisfies the Palais-Smale condition on X ((P S)-condition, for short) if any sequence (u n ) ⊂ X with (I(u n )) bounded and I ′ (u n ) → 0 as n → ∞, possesses a convergent subsequence. By (P S)-sequence for I we understand a sequence (u n ) ⊂ X which satisfies the conditions: (I(u n )) is bounded and I ′ (u n ) → 0 as n → ∞.
The main tools used in proving Theorem 1 are the well known mountain pass theorem and its the symmetric mountain pass theorem. Then I possesses at least dim W − dim Y pairs of nontrivial critical points.

Proof of Theorem 1
To prove Theorem 1 we recall some lemmas presented below.
Lemma 2. Assume that (A 1 ), (F 0 ) and (G 2 ) hold. Then the functional I λ satisfies the Palais-Smale condition on X ((P S)-condition, for short) at any level d.
P r o o f. Let d ∈ R and let (u n ) ⊂ X be (P S) sequence for I λ , then First, we prove that sequence (u n ) is bounded in X. Suppose (u n ) unbounded, we may assume u n → +∞ as n → ∞.
By (2), (A 1 ), (F 0 ) and Proposition 6 we have From (3.2), (F 0 ) and Proposition 6 we obtain Meanwhile, according to (F 0 ), Proposition 4 and Proposition 2 it yields where C r − and C r + are constants of compact embedding X ֒→ L p(x) (Ω). Using (3.1), (3.2), (3.3), (3.4) and (G 2 ) we obtain where d is defined in (3.1). Since p − ≥ r + (u n ) is bounded. Now, with standard arguments, we prove that any (P S) d sequence (u n ) in X has a convergent subsequence. Indeed, the space X is a Banach reflexive space then there exists u ∈ X such that, up to subsequence still denoted by (u n ) and by the Sobolev embedding, we obtain: • u n ⇀ u in X as n → ∞; • u n (x) → u(x) a.e. in Ω as n → ∞; • u n → u in L p(x) (Ω) as n → ∞; and lim n→∞ ∂Ω P r o o f. To demonstrate (3.6), we use Propositions 1-4 we give where C is positive constant. By the compact embedding X ֒→ L p(x) (Ω) and the inequality ||u n | p(x) − |u| p(x) | ≤ |u n − u| p(x) , we obtain |u n − u| p(x) → 0 in L p(x) (Ω) and |u n | p(x) → |u| p(x) . Similar arguments establish (3.7). Now, in virtue of (3.1) and Proposition 8, we have lim sup n→∞ Ω a(|∇u n | p(x) )|∇u n | p(x)−2 ∇u n (∇u n − ∇u)dx Finally, by Lemma 1 u n → u in X.
To finish the proof of the Theorem 1, we check the geometrical conditions of mountain pass Theorem 2 for I λ . Indeed (I 1 ) since the embeddings X ֒→ L i(x) (Ω) (i := p, r, q) and X ֒→ L i(x) (∂Ω) (i := p, q) is are compact, there exist positive constants C i such that (3.8) From (G 0 )-(G 1 ) it follows, for all ε > 0, there exists C ε > 0, such that thus, for u ∈ X with u ≤ 1. By (A 1 ), (3.2), (3.4), (3.8) and (3.9), we have where If ρ = u , we obtain A straightforward computation shows that the maximum of the function ψ is Inserting this into equation (3.11), we find that the right side is zero for So, there exist ρ > 0 and ̺ > 0 such that I λ (u) ≥ ̺ for u = ρ, from which the demonstration of (I 1 ) is completed.
Since I λ (0) = 0, the mountain pass lemma implies the existence of a nontrivial weak solution u 1 with I λ (u 1 ) ≥ ̺.
Hence problem (1.1) has at least one nontrivial weak solution in X.
To complete the proof of the Theorem 1, one must check the conditions of the Theorem 3. So we need some lemmas which we recall below. Remark 2. [30] As the Sobolev space X is a reflexive and separable Banach space, there exist (e n ) n∈N * ⊆ X and (f n ) n∈N * ⊆ X * such that f n (e m ) = δ nm for any n, m ∈ N * and X = span{e n : n ∈ N * }, X * = span{f n : Lemma 3. Assume that (A 0 )-(A 1 ), (F 0 )-(F 1 ) and (G 0 )− (G 1 ) hold. Then there exists λ > 0, k ∈ N and ρ, θ > 0 such that I λ /∂B ρ ∩ X k ≥ θ for all 0 < λ < λ.
P r o o f. Similarly to (3.10), we have Taking ρ = u , we get Next, we take λ = C 1 /C 2 · ρ p + −r + > 0 so that which shows that I verifies the condition (I' 1 ) in Theorem 3.
According to Lemma 2, we also have that I λ satisfies (I' 3 ). Since I λ (0) = 0 and I λ is even, we may apply Theorem 3 to conclude that I λ has infinitely many nontrivial solutions.