Existence and Multiplicity Results for Steklov Problems with p(.)-Growth Conditions

Using variational methods, we prove in different situations the existence and multiplicity of solutions for the following Steklov problem: -div(a(x,∇u))+|u|p(x)-2u=0,inΩ,a(x,∇u).ν=g(x,u),on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\text{ div }(a(x, \nabla u))+|u|^{p(x)-2}u= & {} 0, \quad \text {in }\Omega , \\ a(x, \nabla u).\nu= & {} g(x,u), \quad \text {on } \partial \Omega , \end{aligned}$$\end{document}where Ω⊂RN(N≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N(N \ge 2)$$\end{document} is a bounded domain with smooth boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} and ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} is the unit outward normal vector on ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}. p:Ω¯↦R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p: {\overline{\Omega }} \mapsto {\mathbb {R}}$$\end{document}, a:Ω¯×RN↦RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a: {\overline{\Omega }}\times {\mathbb {R}}^{N} \mapsto {\mathbb {R}}^{N}$$\end{document} and g:∂Ω×R↦R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g: \partial \Omega \times {\mathbb {R}} \mapsto {\mathbb {R}}$$\end{document} are fulfilling appropriate conditions.


Introduction and main results
In this article, we consider the elliptic problem with nonlinear boundary conditions and variable exponents − div(a(x, ∇u)) + |u| p(x)−2 u = 0, in , a(x, ∇u).ν = g(x, u), on ∂ , (1.1) where ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary ∂ , ν is the unit outward normal vector on ∂ and the function involved in this problem will be described in what follows.
p ∈ C( ) are variable exponents and throughout this paper, we denote and A first remark is that hypothesis (H 0 ) is only needed to obtain the multiplicity of solutions. As in [5], we have decided to use this kind of function a satisfying (H 0 )-(H 5 ) because we want to assure a high degree of generality to our work. Here, we invoke the fact that, with appropriate choices of a, we can obtain many types of operators. We give, in the following, two examples of well-known operators which are presented in lots of papers.
The above operator appears in [5] and it is used in the study of two antiplane frictional contact problems of elastic cylinders. Functions fulfilling conditions related to (H 0 )-(H 5 ) are used not only in the framework of the spaces with variable exponents [4,18], but also in the framework of the classical Lebesgue-Sobolev spaces [21] and the anisotropic spaces with variable exponents.
The study of differential and partial differential equation with variable exponent has been received considerable attention in recent years. This importance reflects directly into a various range of applications. There are applications concerning elastic materials [22], image restoration [6], thermorheological and electrorheological fluids [3,19] and mathematical biology [12].
In the case when p(x) = p is a constant and div(a(x, ∇u)) = div(|∇u| p−2 ∇u) with Dirichlet boundary conditions, many authors consider the type problem Ambrosetti and Rabinowitz proposed that the mountain pass theorem in 1973 (see, [1]) critical points theory has became one of the main tools for finding solutions to elliptic problems of variational type. One of the very important hypotheses usually imposed on the nonlinearities is the following Ambrosetti-Rabinowitz type condition [(A-R) type condition for short]: There exists μ > p such that (1.7) Equation (1.7) means that g is p-superlinear at infinity in the sense that lim |t|→+∞ Our purpose of this work is to extend some of the known results with Neuman or Dirichlet boundary conditions on bounded domain (see, [5,7]), and generalize some known results in the Steklov problems (see, [2,20]). We consider the following cases: where λ ∈ R + , r : → R and f : ∂ ×R → R are fulfilling appropriate conditions.
We enumerate now the hypotheses concerning the functions f and F, where for a.e. x ∈ ∂ and all t ∈ R, where q ∈ C + ( ) with q + < p − . (I2) There exist θ > p + and l > 0 such that f satisfies the Ambrosetti-Rabinowitz condition for all |t| > l and a.e.
Our main results in this paper are the proofs of the following theorems, which are based on the different version of mountain pass theorem. The first three theorems concern the case (i) and the rest of theorems deal the case (ii).
Then for any λ > 0 the problem (1.1) in the case (i) possesses a nontrivial weak solutions.
Then, there exists λ 1 > 0, such that for any λ > λ 1 there exists a sequence (u k ) of nontrivial weak solutions for the problem (1.1) in the case (i). Moreover, u k → 0, as k → ∞. This paper consists of four sections. Sect. 1, contains an introduction and our main results. In sect. 2, which has a preliminary character, we state some elementary properties concerning the generalized Lebesgue-Sobolev spaces and embedding results. Here, we also collect the ingredients of our proofs. The proofs of our main results are given as follows: the case (i) is studied in Sects. 3 and 4 is concerned by case (ii). The mountain pass theorem of Ambrosetti and Rabinowitz (see for example Theorem 2.3) is used to show Theorems 1.3 and 1.4. For the proof of Theorem 1.5, we use the version of the symmetric mountain pass theorem (see, Theorem 2.4). Finally, the proof of Theorem 1.6 relies on a classical Weierstrass type theorem (see, Theorem 2.5) and the proof of the last theorem is based on Theorem 1.6 and the previous mountain pass theorem of Ambrosetti and Rabinowitz.

Preliminaries
We first recall some basic facts about the variable exponent Lebesgue-Sobolev.
For p ∈ C + (¯ ), we introduce the variable exponent Lebesgue space which is separable and reflexive Banach space (see, [16]). Let us define the space The properties of W 1, p(x) ( ) and the properties concerning the embedding results are given in the following proposition.
The mapping plays an important role in manipulating the generalized Lebesgue-Sobolev spaces.
For the proofs of Theorems 1.3 and 1.4, we will apply the following mountain pass theorem of Ambrosetti-Rabinowitz.

Theorem 2.3 [14]
Let X endowed with the norm . X , be a Banach space. Assume that φ ∈ C 1 (X ; R) satisfies the Palais-Smale condition. Also, assume that φ has a mountain pass geometry, that is, (i) there exist two constants r > 0 and ρ ∈ R such that φ(u) ≥ ρ if u X = r; (ii) φ(0) < ρ and there exists e ∈ X such that e X > r and φ(e) < ρ.
Then, φ has a critical point u 0 ∈ X such that u 0 = 0 and u 0 = e with critical value where P denotes the class of the paths γ ∈ C([0, 1]; X ) joining 0 to e.
The key argument in the proof of Theorem 1.5 is the following version of the symmetric mountain pass theorem. 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 → ∞.
Finally, we remind the Weierstrass type theorem that will be used in the proof of Theorem 1.6.

Theorem 2.5 ([8])
Assume that X is a reflexive Banach space and the function : X −→ R is coercive and (sequentially) weakly lower semicontinuous on X . Then, is bounded from below on X and attains its infimum on X .

Proofs of Our Main Results in the Case (i)
The energy functional corresponding to problem (1.1) in the case (i) is defined on where dσ is the N − 1 dimensional Hausdorff measure restricted to the boundary ∂ .

Let us recall that a weak solution of problem (1.1) in the case (i) is any
It is known that the weak solutions of problem (1.1) given by (3.2) are exactly the critical points of φ λ, f defined by (3.1).

Proof of Theorem 1.3
We prove now that the mountain pass theorem of Ambrosetti-Rabinowitz (Theorem 2.3) can be applied. We organize this proof as follows: Proof According to the fact that we deduce that for all u ∈ W 1, p(x) ( ), we have Since r − ≤ r + < p ∂ (x) for any x ∈¯ , then by Proposition 2.1, W 1, p(x) ( ) is continuously embedded in L r + (∂ ) and in L r − (∂ ). It follows that there exist two positive constants C 1 and C 2 such that Thus, As p + < r − ≤ r + , the functional h : [0, 1] → R defined by is positive on neighborhood of the origin. So Lemma 3.2 is proved.

Proof of Theorem 1.4
As in the proof of Theorem 1.3, this proof is based on the Theorem 2.3. For deducing that there exists a nontrivial critical point where P denotes the class of the paths γ such that γ ∈ C [0, 1], W 1, p(x) ( ) joining 0 to e, we need to show the following Lemmas. Proof Let c ≥ 0 and (u n ) ⊂ W 1, p(x) ( ) be such that |φ λ, f (u n )| < c and φ λ, f (u n ) → 0 as n → ∞. We first show that (u n ) n is bounded. To do so, we argue by contradiction and we assume that, up to a subsequence, ||u n || → ∞.
Then, using (I2) and (H 5 ), for sufficiently large n we have Using Proposition 2.1 and (I2), we deduce that, for sufficiently large n, Letting n go to infinity and dividing by ||u n || p − in the above inequality, since r + < p − , then we obtain a contradiction. Therefore, (u n ) n is bounded in W 1, p(x) ( ). For a subsequence of (u n ) n , u n u weakly in W 1, p(x) ( ), strongly in L r (x) (∂ ) and in L p(x) ( ). Therefore, φ λ, f (u n ), u n − u → 0, Consequently by Theorem 3.3, u n → u strongly in W 1, p(x) ( ).

Proof of Theorem 1.5
We show now that the symmetric mountain pass theorem (see Theorem 2.4) can be applied. Proof It is clear, by the properties of a(., .) and f , that φ λ f ∈ C 1 , φ λ, f is even and φ λ, f (0) = 0. Using (H 5 ) and (I 1), we have As p(x) ≤ p + , r − ≤ r (x) and q − ≤ q(x) for any x ∈ , we obtain By Proposition 2.1, we deduce that there exist two positive constants C and C such that Proof of Theorem 1.5 By Lemmas 3.7, 3.8 and Theorem 2.4, the problem (1.1) admits a sequence of non-trivial weak solutions (u k ), such that φ λ,0 (u k ) < 0 and lim k→+∞ u k = 0.

Proofs of Our Main Results in Case (ii)
The energy functional corresponding to problem (1.1) the case (ii) is defined on W 1, p(x) ( ) as follows:

Proof of Theorem 1.6
To apply Theorem 2.5, we need the following two Lemmas.
Proof It is known that the functional u −→ [A(x, ∇u) + |u| p(x) p(x) ]dx defined on W 1, p(x) ( ) is weakly lower semicontinous (see, [5,Lemma 5]). At the same time, hypothesis (I5) implies the existence of a positive constant k such that | f (x, t)| ≤ k (1 + |t| p(x)−1 ) for all t ∈ R and a.e. x ∈ ∂ . (4.3) Hence, since the embedding from W 1, p(x) ( ) to L p(x) (∂ ) is compact, standard arguments infer that ψ λ, f is weakly lower semicontinuous for every λ > 0 and the proof of this lemma is complete.  By Proposition 2.2, we have that, for all λ > λ 0 and all u ∈ W 1, p(x) ( ) with u = r < 1, Then, since p + < s − and choose small enough such that ( 1 p + − λc 2 ) > 0, for r sufficiently small, there exists ρ > 0 such that ψ λ, f ≥ ρ > 0 for all λ > λ 0 > 0 and all u ∈ W 1, p(x) ( ) with u = r < min(1, u 1 ). Thus, we can apply now Theorem 2.3 to find a second critical point u 2 ∈ W 1, p(x) ( ) with In this way, we have obtained a second nontrivial weak solution of the problem (1.1) in the case (ii) and the proof of Theorem 1.7 is complete.