Existence of Infinitely Many Solutions for a Steklov Problem Involving the P(x)-laplace Operator

In this article, we study the nonlinear Steklov boundary-value problem ∆ p(x) u = |u| p(x)−2 u in Ω, |∇u| p(x)−2 ∂u ∂ν = f (x, u) on ∂Ω. We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

The purpose of this article is to study the existence and multiplicity of solutions for the Steklov problem involving the p(x)-Laplacian, ∆ p(x) u = |u| p(x)−2 u in Ω, |∇u| p(x)−2 ∂u ∂ν = f (x, u) on ∂Ω. (1.1) where Ω ⊂ R N (N ≥ 2) is a bounded smooth domain, ∂u ∂ν is the outer unit normal derivative on ∂Ω, p is a continuous function on Ω with N < p − := inf x∈Ω p(x) ≤ p + := sup x∈Ω p(x) < +∞ and f : ∂Ω × R → R is a continuous function.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.Many authors have studied the inhomogeneous Steklov problems involving the p-Laplacian [14].The authors have studied this class of inhomogeneous Steklov problems in the cases of p(x) ≡ p = 2 and of p(x) ≡ p > 1, respectively.From now, we put X = W 1,p(x) (Ω) and w := 2π N/2 NΓ( N

)
the measure of the N-dimensional unit ball.
The main results of this paper are as follows.
Example 1.4.An example of functions satisfying the assumptions of Theorem 1.2 where h ∈ C(Ω) with min x∈Ω h(x) ≥ h 0 , z ∈ C(Ω) with min x∈Ω z(x) > 1 and q ∈ C(Ω) with p − − α ≤ q(x) ≤ p − for all x ∈ Ω.Note that in this occasion we can choose Existence of infinitely many solutions for boundary value problems have received a great deal of interest in recent years, see, for instance, the paper [2,5] and references therein.In [1] we have considered the existence and multiplicity of solutions for the Steklov problem involving the p(x)-Laplacian of the type Under the following assumptions of the function f , we have established the existence of at least three solutions of this problem.
This article is organized as follows.First, we will introduce some basic preliminary results and lemmas in Section 2. In Section 3, we will give the proofs of our main results.

Preliminaries
For completeness, we first recall some facts on the variable exponent spaces L p(x) (Ω) and W k,p(x) (Ω).For more details, see [9,10].Suppose that Ω is a bounded open domain of R N with smooth boundary ∂Ω and p ∈ C + (Ω) where Denote by p − := inf x∈Ω p(x) and p + := sup x∈Ω p(x).Define the variable exponent Lebesgue space L p(x) (Ω) by Define the variable exponent Sobolev space W 1,p(x) (Ω) by We refer the reader to [9,10] for the basic properties of the variable exponent Lebesgue and Sobolev spaces.
Infinitely many solutions for a Steklov problem 5 2. If there exist a sequence {r k } ⊂ (inf X φ, +∞) with r k → (inf X φ) + and a sequence {u k } ⊂ X such that for each k the conditions (2.1) and (2.2) are satisfied, and in addition, every global minimizer of φ is not a local minimizer of φ + ψ, then there exists a sequence {v k } of pairwise distinct local minimizers of φ + ψ such that lim k→+∞ φ(v k ) = inf X φ, and {v k } weakly converges to a global minimizer of φ.
Definition 2.7.We say that u ∈ X is a weak solution of (1.1) if For each u ∈ X, we define where F(x, t) = t 0 f (x, s) ds.Then we have Then it is easy to see that φ, ψ ∈ C 1 (X, R) and u ∈ X is a weak solution of (1.1) if and only if u is a critical point of the functional J.
Notice that φ is convex and continuous functional so it is a weakly lower semi-continuous.Since the embedding X → C(Ω) is compact, we can see that ψ : X → R is sequentially weakly lower semi-continuous.

Proof of main results
For the proof of Theorems 1.1 and 1.2, we will use Lemma 2.6.We start with the following lemmas.
then φ is coercive.The proof is completed.

So we have inf
Proof.Let r ≥ 1 p + and u ∈ X be such that φ(u) < r.When u ≥ 1, by (3.1), we obtain which implies that u < rp + Proof of Theorem 1.1.We use Lemma 2.6(1) to prove Theorem 1.1.
By condition (2), we have max with Without loss of generality, we may assume that η k ≥ max(γ, 1).In view of condition (1), we choose for all k > k 1 .For each k > k 1 , using (3.6), we have Infinitely many solutions for a Steklov problem For each v ∈ φ −1 ((−∞, r k )), we can easily see that for each x ∈ ∂Ω By condition (4), there exists a Therefore, using (3.9), we obtain sup k for a.e.x ∈ ∂Ω.Now we consider a function w k ∈ X defined by w k (x) = ξ k .Without loss of generality, we may assume that ξ k ≥ 1.So we have if u is weak solution of the problem (S + ), then one has for all v ∈ X. Taking v = u − in (3.11) shows that u − = 0, so u − = 0. Obviously, u is a non-negative solution of (1.1) in X.This completes the proof.
We put r k = 1  We can easily get (2.1) and (2.2) using the same method as in the proof of Theorem 1.1.In view of condition (3), we can find a sequence {ξ k } k∈N ⊂ R such that lim k→+∞ ξ k = 0 + and Infinitely many solutions for a Steklov problem 9 F(x, ξ k ) ≥ h 0 ξ p − k for a.e.x ∈ ∂Ω.If we take w k = ξ k , of course the sequence {w k } strongly converges to 0 in X and φ(w k ) + ψ(w k ) < 0 for all k ∈ N. Since φ(0) + ψ(0) = 0, this means that 0 is not a local minimum of φ + ψ.
So, since 0 is the only global minimum of φ.Lemma 2.6(2) ensures that there exists a sequence {v k } of pairwise distinct local minimizers of φ + ψ such that lim k→+∞ φ(v k ) = 0. Using the same method as in the proof of Theorem 1.1, we can get that each weak solution of problem (1.1) is non-negative.This completes the proof.