Existence of solutions for a Steklov problem involving the p ( x )-Laplacian

. The operator ∆p(x) := div(|∇u| ∇u) is the p(x)Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant). Nonlinear boundary value problems with variable exponent has been received considerable attention in recent years. This is partly due to their frequent appearance in applications such as the modeling of electro-rheological fluids [1,16] and image processing [3], but these problems are very interesting from a purely mathematical point of view as well. Many results have been obtained on this kind of problems; see for example [4,5,6,17,18]. In [4], the authors have studied the


Introduction and Main Results
The purpose of this paper is to study the existence of solutions for the following nonlinear boundary value problem involving the p(x)-Laplacian in Ω, |∇u| p(x)−2 ∂u ∂ν = λ|u| q(x)−2 u on ∂Ω, where Ω is a bounded domain in R N (N ≥ 2) with smooth boundary ∂Ω, ν is the unit outward normal to ∂Ω, λ is a positive number and p(.), q(.) The operator ∆ p(x) := div(|∇u| p(x)−2 ∇u) is the p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant).
Nonlinear boundary value problems with variable exponent has been received considerable attention in recent years.This is partly due to their frequent appearance in applications such as the modeling of electro-rheological fluids [1,16] and image processing [3], but these problems are very interesting from a purely mathematical point of view as well.Many results have been obtained on this kind of problems; see for example [4,5,6,17,18].In [4], the authors have studied the case p(x) = q(x) for all x ∈ Ω, they proved that the existence of infinitely many eigenvalue sequences.Unlike the p-Laplacian case, for a variable exponent p(x) ( ≡ constant), there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions.Finally, they presented some sufficient conditions for the infimum of all eigenvalues is zero and positive, respectively.
Throughout this paper, we denote by h Our main results in this paper are the proofs of the two following theorems, which are based on the Mountain Pass Theorem.
This paper consists of four sections.Section 1 contains an introduction and the main results.In Section 2, we state some elementary properties concerning the generalized Lebesgue-Sobolev spaces and an embedding results.The proofs of Theorem 1.1 and Theorem 1.2 are given respectively in Section 3 and Section 4.

Preliminaries
We first recall some basic facts about the variable exponent Lebesgue-Sobolev.For p ∈ C + ( Ω), we introduce the variable exponent Lebesgue space which is separable and reflexive Banach space (see [15]).Let us define the space Proposition 2.1.[7,12,13,15] (1) W 1,p(x) 0 (Ω) is separable reflexive Banach space; (2) If q ∈ C + ( Ω) and q(x) < p * (x) for any x ∈ Ω, then the embedding from W 1,p(x) (Ω) to L q(x) (Ω) is compact and continuous; (3) If q ∈ C + ( Ω) and q(x) < p * ∂ (x) for any x ∈ Ω, then the embedding from An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping ρ defined by The key argument in the proof of Theorem 1.1 is the following version of the Symmetric Mountain Pass Theorem (see [14]).The energy functional corresponding to problem (1.1) is defined as where dσ is the N − 1 dimensional Hausdorff measure.Standard arguments imply that φ λ ∈ C 1 (W 1,p(x) (Ω), R) and for any u, v ∈ W 1,p(x) (Ω).Thus the weak solutions of problem (1.1) are exactly the critical points of φ λ .
We show now that the Symmetric Mountain Pass Theorem can be applied in this case.Lemma 3.2.Let p, q ∈ C + ( Ω). Assume that q + < p − .Then the functional φ λ is even, bounded from below, satisfies the Palais-Smale (P S) condition and φ λ (0) = 0.
Proof: It is clear that φ λ is even and φ λ (0) = 0.According to the fact that we deduce that for all u ∈ W 1,p(x) (Ω), we have Since q + < p − < p * ∂ (x) for any x ∈ Ω, then by Proposition 2.1, W 1,p(x) (Ω) is continuously embedded in L q + (∂Ω) and in L q − (∂Ω).It follows that there exist two positive constants C 1 and C 2 such that Relations (3.1) and (3.2) imply Using (3.3) and Proposition 2.2, we have As q + < p − , φ λ is bounded from below and coercive.It remains to show that the functional φ λ satisfies the (PS) condition to complete the proof.
According to the fact that the operator A satisfies condition (S + )(see [8,9,11]), we deduce that u k → u in W 1,p(x) (Ω), this completes the proof.✷ Lemma 3.3.Let p, q ∈ C + ( Ω). Assume that q + < p − .Then for each k ∈ N * , there exists an We show now that for any k ∈ N * , there exists Indeed, for 0 < t ≤ 1, we have Let c = min v∈F k ∩S ∂Ω |v| q(x) dσ > 0, we may choose t k ∈]0, 1] which is small enough such that 1 p − − cλ q + t p − −q + < 0. For the proof of Theorem 1.2, we want to construct a mountain geometry.
Proof: Since q + < p * ∂ (x) for all Ω, similar arguments as those used in the proof of Lemma 3.2 give the following inequalities Thus As p + < q − ≤ q + , the functional h : [0, 1] → R defined by is positive on neighborhood of the origin.So the Lemma 4.2 is proved.✷ Lemma 4.2.Let p, q ∈ C + ( Ω). Assume that p + < q − ≤ q + < p * ∂ (x) for all x ∈ Ω.Then there exists e ∈ W 1,p(x) (Ω) with e > η such that φ λ (e) < 0; where η is given in Lemma 4.1.
Since p + < q − , this contradicts the fact that p − > 1.So, the sequence (u k ) is bounded in W 1,p(x) (Ω) and similar arguments as those used in the proof of Lemma  This completes the proof.✷

Theorem 3 . 1 .
Let E be an infinite dimensional Banach space and I ∈ C 1 (E, R) satisfy the following two assumptions (A 1 ).I(u) is even, bounded below; I(0) = 0 and I(u) satisfies the Palais-Smale condition (PS);(A 1 ).For each k ∈ N, there exists an A k ∈ Γ k such that supu∈A k I(u) < 0.Then I(u) admits a sequence of critical points u k such that I(u k ) < 0; u k = 0 and u k → 0, as k → ∞.Where Γ k denote the family of closed symmetric subsets A of E such that 0 ∈ A and γ(A) ≥ k with γ(A) := inf{k ∈ N; ∃h : A → R k \{0} such that h is continuous and odd} is the genus of A.