Solutions for Steklov boundary value problems involving p ( x )-Laplace operators

In this paper we study the nonlinear Steklov boundary value problem of the following form: (S) { ∆p(x)u = |u| p(x)−2u in Ω, |∇u|p(x)−2 ∂u ∂ν = λf(x, u) on ∂Ω. Using the variational method, under appropriate assumptions on f , we establish the existence of at least three solutions of this problem.


Introduction
In recent years, the study of differential equations and variational problems with p(x) growth conditions has been an interesting topic.We refer to [6,7,8,10,13,14,15] for the p(x)-Laplacian equations.This paper is motivated by recent advances in mathematical modeling of non-Newtonian fluids and elastic mechanics, in particular, the electro-rheological fluids (smart fluids).This important class of fluids is characterized by the change of viscosity which is not easy and which depends on the electric field.These fluids, which are known under the name ER fluids, have many applications in elastic mechanics, fluid dynamics etc..For more information, the reader can refer to [12,16,17].
These physical problems was facilitated by the development of Lebesgue and Sobolev spaces with variable exponent.The existence of solutions of p(x)-Laplacian problems has been studied by several authors (see [3,6,8,9,13,14]).
2000 Mathematics Subject Classification: 35J48, 35J60, 35J66 164 M. Allaoui, A. R. El Amrouss Consider the following nonlinear and inhomogeneous Steklov boundary problem, ∂ν is the outer unit normal derivative on ∂Ω, λ > 0 is a real number, p is a continuous function on Ω with 1 < p − := inf x∈Ω p(x) ≤ p + := sup x∈Ω p(x) < +∞.The main interest in studying such problems arises from the presence of the p(x)-Laplace operator div(|∇u| p(x)−2 ∇u), which is a generalization of the classical p-Laplace operator div(|∇u| p−2 ∇u) obtained in the case when p is a positive constant.
Using the three critical point theorem due to Ricceri, under the above assumptions on f , we establish the existence of at least three solutions of this problem.
Theorem 1.1.If the function f satisfies (f 1 ) − (f 3 ) and the function p satisfies p − > α + , then there exist an open interval Λ ⊂ (0, ∞) and a positive real number ρ such that for each λ ∈ Λ, (S) has at least three solutions whose norms are less than ρ.
This paper is divided into three sections.In Section 2, we recall some basic facts about the variable exponent Lebesgue and Sobolev spaces, and recall B. Ricceri's three-critical-points theorem.In Section 3, we give the proof of theorem1.1.

Preliminaries
To guarantee completeness of this paper, we first recall some facts on variable exponent spaces L p(x) (Ω) and W 1,p(x) (Ω).For more details, see [2,4,5].Define the variable exponent Lebesgue space L p(x) (Ω) by Define the variable exponent Sobolev space W 1,p(x) (Ω) by with the norm We refer to [4,5,6] for the basic propreties of the variable exponent Lebesgue and Sobolev spaces.
Lemma 2.2.(see [5]) Hölder inequality holds, namely where we have Lemma 2.4.(see [4]) Assume that the boundary of Ω possesses the cone property and p ∈ C(Ω) and 1 ≤ q(x) < p * (x) for x ∈ Ω, then there is a compact embedding where Lemma 2.5.(see [5]) If f : Ω × R → R is a carathéodory function and is a continuous and bounded operator.
If u, ϕ ∈ X, then by condition (2.1) and the embedding theorem (lemma2.7),we have u, ϕ ∈ L α(x) (∂Ω).Then there is some constant C 1 such that By (f 1 ) and Young ′ s inequality, we have (2.4) Using the inequality which implies that for |t| ≤ 1, Notice that the right hand side of the above inequality is independent of t and integrable on ∂Ω, then by the Lebesgue dominated convergence theorem, we have Obviously the operator Dψ(u, ϕ) is a linear operator for a given u.We know that the Nemytskii operator α(x)−1 (∂Ω).Then by (2.3) and (2.5), we have Then Dψ(u, ϕ) is a linear bounded functional, therefore the Gâteaux derivative of the linear bounded functional ψ(u) exists and We will prove that ψ ′ : X → X * is completely continuous.For u, v, ϕ ∈ X, from (2.5) and (2.6), we obtain Existence and multiplicity of solutions .

Proof of Theorem 1.1
To prove our result we use lemma 2.8.It is well known that φ is a continuous convex functional, then it is weakly lower semicontinuous and its inverse derivative is continuous, from theorem 2.9 the precondition of lemma 2.8 is satisfied.In following we must verify that the conditions (i), (ii) and (iii) in lemma 2.8 are fulfilled.
In view of (f 1 ), if we put where A is a positive constant, then we have F (x, t) ≤ C 5 |t| q − , ∀t ∈ R uniformly for x ∈ ∂Ω.