On the Superlinear Steklov Problem Involving the P(x)-laplacian

This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian Steklov problem without the well-known Ambrosetti–Rabinowitz type growth conditions. By means of critical point theorems with Cerami condition, under weaker conditions, existence and multiplicity results of the solutions are proved.


Introduction
The study of differential equations and variational problems with nonstandard p(x)-growth conditions has received more and more interest in recent years.The specific attention accorded to such kinds of problems is due to their applications in mathematical physics.More precisely, such equations are used to model phenomena which arise in elastic mechanics or electrorheological fluids, we can refer the reader to [15].This kind of problems has been the subject of a sizeable literature and many results have been obtained, see for example [1,6,8,17] and references therein.

A. Ayoujil
Define the family of functions where p + := sup x∈Ω p(x).
Noticing that when p(x) = p is a constant, F = { f (x, t)t − pF(x, t)} consists of only one element.
We limit ourselves to the subcritical case, i.e. we assume that (f 1 ) there exist c > 0 and q ∈ C + (∂Ω) with q(x) < p ∂ (x) for each x ∈ ∂Ω, such that for each (x, t) ∈ ∂Ω × R, where Problems like (1.1) have been largely considered in the literature in the recent years.In [8], the authors have studied the case f (x, u) = λ|u| p(x)−2 u.They proved the existence of infinitely many eigenvalue sequences and that unlike the p-Laplacian case, there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions.Moreover, they presented some sufficient conditions for the infimum of all eigenvalues to be zero and positive, respectively.In [14], the authors have studied the inhomogeneous Steklov problems involving the p-Laplacian.They studied this class of inhomogeneous Steklov problems in the cases of p(x) = p = 2 and of p(x) = p > 1, respectively.Recently, in [1] the authors obtained results on existence and multiplicity of solutions for problem (1.1) in the case q − > p + , under (f 1 ) and the following conditions: (AR) There exists M > 0 and θ > p + such that Generally, to show the existence of solutions for problems which are superlinear, it is essential to assume the superquadraticity condition (AR), which is known as Ambrosetti-Rabinowitz's type condition [2].It is well known that the main aim of using (AR) is to ensure the boundedness of the Palais-Smale type sequences of the corresponding functional.But this condition is very restrictive eliminating many nonlinearities.In fact, there are many functions which are superlinear but do not satisfy (AR), see the example in Remark 1.1 below.
As far as we are aware, elliptic problems like (1.1) involving the p(x)-Laplacian operator without the (AR) type condition, have not yet been studied.That is why, at our best knowledge, the present paper is a first contribution in this direction.In the present paper, we do not On the superlinear Steklov problem 3 use (AR) and we know that without (AR) it becomes a very difficult task to get the boundedness.So, using a weaker assumption (g) below instead of (AR) and some variant min-max theorem, which will be reminded in Section 2, we overcome these difficulties.
At first, we will show the existence of a nontrivial weak solution by means of a version of the mountain pass theorem with the Cerami condition [3,7].As we will show later, the hypotheses (f 1 ) and (f 2 ) imply the mountain pass geometry for the functional corresponding to problem (1.1).To insure the Cerami condition, we introduce some natural growth hypotheses on the nonlinear term in (1.1).More precisely, we assume that the following hold: Remark 1.1.Obviously, (f 4 ) can be derived from (AR).However, when p(x) ≡ 2, δ = 1 it is easy to see that function does not satisfy (AR), while it satisfies the aforementioned conditions.
Secondly, we will prove under some symmetry condition on the function f that the problem (1.1) possesses infinitely many nontrivial weak solutions.The proof is based on a variant of the fountain theorem [13].
By a weak solution to problem (1.1) we understand a function u ∈ where dσ is the measure on the boundary.
The energy functional corresponding to problem (1.1) is defined as I : X → R, where ) dx and Ψ(u) = ∂Ω F(x, u) dσ.Let us note that under the hypothesis (f 1 ), the functional I is well defined and of class C 1 and the Fréchet derivative is given by for any u, v ∈ X.Moreover, the critical points of I are weak solutions of (1.1).
Our main results are stated as follows.
The present article is composed of three sections.Section 2 contains some useful results on Sobolev spaces with variable exponents.In particular, we recall a weighted variable exponent Sobolev trace compact embedding theorem and some min-max theorems like mountain pass theorem and fountain theorem with the Cerami condition that will be useful later.The proofs of the main results are given in Section 3.
Throughout the sequel, the letters c, c i , i = 1, 2, . . ., denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process.

Preliminaries
To discuss problem (1.1), we need some theory of variable exponent Lebesgue-Sobolev spaces.For convenience, we only recall some basic facts which will be used later.For details, we refer to [9,10,12].
For p ∈ C + (Ω), we designate the variable exponent Lebesgue space by equipped with the so-called Luxemburg norm is a continuous and bounded operator.
As in the constant exponent case, the generalized Lebesgue-Sobolev space W 1,p(x) (Ω) is defined as With such norms, L p(x) (Ω) and W 1,p(x) (Ω) are separable, reflexive and uniformly convex Banach spaces.
On the superlinear Steklov problem 5. Then the following statements are equivalent to each other.
Recall the following embedding theorem.
Next we give the definition of the Cerami condition which introduced by G. Cerami [4].Definition 2.4.Let X be a Banach space and I ∈ C 1 (E, R).Given c ∈ R, we say that I satisfies the Cerami c condition (we denote condition (C c )), if (C 1 ) any bounded sequence (u n ) ⊂ E such that I(u n ) → c and I (u n ) → 0 has a convergent subsequence; (C 2 ) there exist constants α, r, β > 0 such that

A. Ayoujil
If I ∈ C 1 (E, R) satisfies condition (C c ) for every c ∈ R, we say that I satisfies condition (C).
Note that condition (C) is weaker than the (PS) condition.However, it was shown in [3,5] that from condition (C) it can obtain a deformation lemma, which is fundamental in order to get some min-max theorems.More precisely, let us recall the version of the mountain pass lemma with Cerami condition which will be used in the sequel.Theorem 2.5 (See [3,7]).Let X a Banach space, I ∈ C 1 (E, R), e ∈ X and r > 0 be such that e > r and inf where Then c is a critical value of I.
We need the following lemma.
Lemma 2.7.For α ∈ C + (∂Ω), α(x) < p ∂ (x) for any x ∈ ∂Ω, define Then lim On the superlinear Steklov problem 7 Proof.It is clear that the sequence (β k ) is nonincreasing and positive, so Passing if necessary to a subsequence, there exists a subsequence, still noted by (u k ), such that (u k ) converges weakly to u in X.
On the other hand, for every j ∈ N, Thus, u = 0.According to Theorem 2.3, there is a compact embedding of X into L α(x) (∂Ω), which assures that (u k ) converges strongly to 0 in L α(x) (∂Ω) and finally that l = 0.

Proofs of main results
First of all, we start with the following compactness result which plays the most important role.
Proof.First, we show that I satisfies the condition (C 1 ).Let (u n ) ⊂ E be bounded such that Passing to a subsequence if necessary, still denoted by (u n ), we may assume that u n u in X.In view of (f 1 ), Ψ : . By the fact that Φ is a homeomorphism in view of Proposition 2.5, we obtain u n → u in X.
Now check that I satisfies the condition (C 2 ) too.Arguing by contradiction, let us suppose that there exist c ∈ R and (u n ) ⊂ E satisfying Let Put v n = u n u n , then v n = 1.Up to subsequences, for some v ∈ E, we have v n (x) → v(x) a.e. in Ω.
If v = 0, as in [11], we can define a sequence (t n ) ⊂ R, such that I(tz n ).
By Lemma 3.1, I satisfies conditions (C) in X.To apply Theorem 2.5, we will show that I possesses the mountain pass geometry.First, we claim that there exist µ, ν > 0 such that I(u) ≥ ν, for u ∈ X with u = µ.
Therefore, in view (3.13), for u sufficiently small we get As q − > p + , by the standard argument, our claim follows.Next, we affirm that there exists e ∈ X \ B µ (0) such that I(e) < 0. (3.14)