Existence and multiplicity of solutions for p(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x)$\end{document}-Laplacian problem with Steklov boundary condition

We study the existence and multiplicity of weak solutions for an elliptic problem involving p(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x)$\end{document}-Laplacian operator under Steklov boundary condition. The approach is based on variational methods.


Introduction
Steklov conditions are considered a more "realistic" description of the interactions at the boundary of a physical system. For example, the heat flow through the surface of a body generally depends on the value of the temperature at the surface itself (see [2,4,11,14] and the references therein for some kinds of Steklov problems).
Recently, Afrouzi et al. [1] studied the existence of multiple solutions of the following Steklov problem involving p(x)-Laplacian operator: where ⊂ R N , N ≥ 2 is a bounded smooth domain, λ is a positive parameter, f : ∂ ×R → R is a Carathéodory function with a growth condition, and a ∈ L ∞ ( ). Also, the existence of at least one positive radial solution belonging to the space W has been proved [21], where B is the unit ball centered at the origin in R N , N ≥ 3, p, q, r ∈ C + (B), R is a positive radial function that satisfies the suitable conditions and a(x) = θ |x| and b(x) = ξ |x| , in which θ , ξ ∈ L ∞ (0, 1) such that θ is a positive nonconstant radially nondecreasing function and ξ is a nonnegative radially nonincreasing function (see [17][18][19][20][22][23][24] and the references therein).
Motivated by their works, here we are interested in finding enough conditions for the existence and multiplicity of weak solutions to the following Steklov p(x)-Laplacian problem: is a Carathéodory function with the following conditions: for (x, s) ∈ × R. And g : × R − → R is a Carathéodory function with the following growth condition: We recall that f : The definition of the weak solution of problem (1.1) is as follows.
Definition 1. 1 We say that the function u ∈ W 1,p(x) ( ) is a weak solution of problem (1.1) if Remark 1.1 An interested reader may study the problem in the Orlicz-Sobolev spaces or on the Heisenberg groups (see [12,13,25,26] and the references therein for details of these spaces).
One of the main results of this paper is as follows.
problem (1.1) admits at least one nontrivial weak solution.
Subsequently, by Theorem 4.1 and Theorem 4.2, we present the existence of two and three weak solutions of problem (1.1), respectively.
The rest of the paper is organized as follows: In Sect. 2, some preliminaries and basic facts are recalled and the function space is introduced. Also some critical point theorems are recalled, and we use them for the main results. In Sect. 3, the existence of at least one weak solution for problem (1.1) is proved. Finally, in Sect. 4 the existence of multiple weak solutions for problem (1.1) is proved.

Function spaces and critical point theorems
We suppose that p ∈ C( ) satisfies the following condition: For any u ∈ L p(x) ( ) and v ∈ L p (x) ( ), where L p (x) ( ) is the conjugate space of L p(x) ( ), the Hölder type inequality holds true. Also, for u ∈ L p(x) (∂ ), we put Following the authors of paper [21], for any κ > 0, we put and for r ∈ C + ( ). The following proposition is well known in Lebesgue spaces with variational exponent (for instance, see [15,Proposition 2.7]).
We denote the variable exponent Sobolev space W 1,p(x) ( ) by As pointed out in [10,16], W 1,p(x) ( ) is continuously embedded in W 1,p -( ), and since p -> N , W 1,p -( ) is compactly embedded in C 0 ( ). Thus, W 1,p(x) ( ) is compactly embedded in C 0 ( ). So, in particular, there exists a positive constant m > 0 such that for each u ∈ W 1,p(x) ( ). When is convex, an explicit upper bound for the constant m (see [8]) is as follows: It is well known that, in view of (2.1), both L p(x) ( ) and W 1,p(x) ( ) are separable reflexive and uniformly convex Banach spaces [10].
Proof Since ess inf c > 0, so there exists 0 < δ < 1 such that δ < c(x). Using Proposition 2.1 and the hypothesis c ∈ L ∞ ( ), we gain and Bearing in mind the following elementary inequality: for all q > 0, there exists C q > 0 such that It is enough to put k = δ The growth condition (F0) gives the following estimate: where M = a γ -mγ | |. The global Ambrosetti-Rabinowitz condition (AR) for g : × R − → R is as follows: There are constants μ > p + , R > 0 such that for all x ∈ ∂ and |s| > R, where G(x, t) := t 0 g(x, s) ds. Definition 2.1 Let and be two continuously Gâteaux differentiable functionals defined on a real Banach space X and fix r ∈ R. The functional I := -is said to verify the Palais-Smale condition cut of upper at r (in short (P.S.) [r] ) if any sequence {u n } n∈N ∈ X such that • I(u n ) is bounded; • lim n→+∞ I (u n ) X * = 0; • (u n ) < r for each n ∈ N; has a convergent subsequence.
The following is one of the main tools of the next section.
) Let X be a real Banach space and , : X − → R be two continuously Gâteaux differentiable functionals such that inf x∈X = (0) = (0) = 0. Assume that there exist positive constants r ∈ R and x ∈ X with 0 < (x) < r such that

Theorem 2.2 ([5]
) Let X be a real Banach space, , : X → R be two continuously Gâteaux differentiable functionals such that is bounded from below and (0) = (0) = 0. Fix r > 0 and assume that, for each the functional I λ :=λ satisfies the Palais-Smale condition and it is unbounded from below. Then, for each the functional I λ admits two distinct critical points.
Finally, we recall the following tool, which is in a convenient form.

Theorem 2.3 ([7]
) Let X be a reflexive real Banach space, : X → R be a coercive, continuously Gâteaux differentiable, and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on X * , : X → R be a continuously Gâteaux differentiable whose Gâteaux derivative is compact such that inf X = (0) = (0) = 0.
Assume that there exist r > 0 andx ∈ X, with r < (x), such that sup (x)<r (x) [, the functional I λ :=λ is coercive. Then, for each λ ∈ r , the functionalλ has at least three distinct critical points in X.

Existence of weak solutions
In this section we deal with the existence of one weak solution for problem (1.1). In fact, we prove the first result of the paper, Theorem 1.1, as follows.
Proof We apply Theorem 2.1. To this end, for each u ∈ W 1,p(x) ( ), let the functionals , : for u ∈ W 1,p(x) ( ). So, weak solutions of (1.1) are exactly the critical points of I λ . The functionals and satisfy the regularity assumptions of Theorem 2.1. Moreover, is sequentially weakly lower semicontinuous and its inverse derivative is continuous (since it is a continuous convex functional). From condition (F1) it is clear that F(x, u) ≤ 0, and thanks to Remark 2.1 and inequality (2.2), one has Also, by standard arguments, we have that is Gâteaux differentiable, and its Gâteaux derivative at the point u ∈ W 1,p(x) ( ) is the functional (u) given by (u)v = |∇u| p(x)-2 ∇u∇v + c(x)|u| p(x)-2 uv dxf (x, u)v dx for every v ∈ W 1,p(x) ( ). On the other hand, the functional is well defined, continuously Gâteaux differentiable with compact derivative, whose Gâteaux derivative at the point u ∈