Existence and multiplicity results for elliptic problems with Nonlinear Boundary Conditions and variable exponents

By applying the Ricceri’s three critical points theorem, we show the existence of at least three solutions to the following elleptic problem: −div[a(x,∇u)] + |u|u = λf(x, u), in Ω, a(x,∇u).ν = μg(x, u), on ∂Ω, where λ, μ ∈ R+, Ω ⊂ R (N ≥ 2) is a bounded domain of smooth boundary ∂Ω and ν is the outward normal vector on ∂Ω. p : Ω 7→ R, a : Ω × R 7→ R , f : Ω× R 7→ R and g : ∂Ω× R 7→ R are fulfilling appropriate conditions.

x ∈ Ω and all s 1 , s 2 ∈ R N .With equality if and only if s 1 = s 2 .
(H 5 ) The inequalities |s| p(x) ≤ a(x, s)s ≤ p(x)A(x, s) hold for a.e.x ∈ Ω and all s ∈ R N .
A first remark is that hypothesis (H 0 ) is only needed to obtain the multiplicity of solutions.As in [6], 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 present in lots of papers.
The energy functional corresponding to problem (1.1) is defined on W 1,p(x) (Ω) as Existence and multiplicity results...

123
where where )ds, and dσ is the N − 1 dimensional Hausdorff measure.Let us recall that a weak solution of (1.1) is any 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 [25], image restoration [7], thermorheological and electrorheological fluids [2,21] and mathematical biology [10].
Ricceri's three critical points theorem is a powerful tool to study boundary problem of differential equation (see, for example, [1,3,4,5]).Particularly, Mihailescu [17] use three critical points theorem of Ricceri [19] study a particular p(x)-Laplacian equation.He proved existence of three solutions for the problem.Liu [16] study the solutions of the general p(x)-Laplacian equations with Neumann or Dirichlet boundary condition on a bounded domain, and obtain three solutions under appropriate hypotheses.Shi [22] generalizes the corresponding result of [17].The multiple solutions of p(x)-biharmonic equation under sublinear condition has been studied in [15] by L. Li, L. Ding and W.W. Pan.To our knowledge, there is no result of multiple solutions of elliptic problems with nonlinear boundary conditions and variable exponents.
We enumerate the hypotheses concerning the functions f, F and g.
(I1) For t ∈ C(Ω) and t(x) < p * (x) for all x ∈ Ω, we have (I2) There exist positive constant c 1 such that F (x, s) > 0 for a.e.x ∈ Ω and all s ∈]0, c 1 ]; (I3) there exist p 1 (x) ∈ C(Ω) and p (I4) There exist positive constant c 2 and a function γ(x) ) for a.e.x ∈ Ω and all s ∈ R; (I5) For p 2 ∈ C(Ω) and p 2 (x) < p ∂ (x) for all x ∈ Ω, we have sup This article is divided into three sections.In Section 2, we recall some basic facts about the variable exponent Lebesgue and Sobolev spaces at first and we recall B. Ricceri's three critical points theorem (Theorem 2.3).In the third section, we prove the following theorem which is the main result of this paper.
In the sequel, we recall the revised form of Ricceri's three critical points theorem [20, Theorem 1] and [18, Proposition 3.1].

Theorem 2.3 ( [20, Theorem 1]
).Let X be a reflexive real Banach space.Φ : X → R is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X ′ , where X ′ is the dual of X, and Φ is bounded on each bounded subset of X; Ψ : X → R is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; I ⊆ R is an interval.Assume that for all λ ∈ I, and that there exists h ∈ R such that Then, there exists an open interval Λ ⊆ I and a positive real number ρ with the following property: for every λ ∈ Λ and every C 1 functional J : X → R with compact derivative, there exists δ > 0 such that, for each µ ∈ [0, δ] the equation has at least three solutions in X whose norms are less than ω.
Proposition 2.4 ( [18, Proposition 3.1]).Let X be a non-empty set and Φ, Ψ two real functions on X. Assume that there are r > 0 and x 0 , x 1 ∈ X such that Then, for each h satisfying In our work, we designate by X the Sobolev space with variable exponent W 1,p(x) (Ω).

Proof of main result
The operator Φ is well defined and of class C 1 (see [6]).The Fréchet derivative of Φ is the operator Φ ′ : X → X ′ defined as We start by proving some properties of the operator Φ ′ .Theorem 3.1.Suppose that the mapping a satisfies (H 0 )-(H 5 ).Then the following statements holds.
Proof: (1) Since Φ ′ is the Fréchet derivative of Φ, it follows that Φ ′ is continuous.Using (H 4 ) and the elementary inequalities [23] which means that Φ ′ is strictly monotone.

127
(2) Let (u n ) n be a sequence of X such that u n ⇀ u weakly in X as n → +∞ and lim sup n→+∞ Using the compact embedding W 1,p(x) (Ω) ֒→ L p(x) (Ω), we have The following theorem assure that u n → u strongly in W 1,p(x) (Ω) as n → +∞.
(3) Note that the strict monotonicity of Φ ′ implies its injectivity.Moreover, Φ ′ is a coercive operator.Indeed, using (H 5 ), Proposition2.2and since p − − 1 > 0, for each u ∈ X such that u ≥ 1 we have Consequently, the operator Φ ′ is a surjection and admits an inverse mapping.It suffices then to show the continuity of Φ By the coercivity of Φ ′ , one deducts that the sequence (u n ) is bounded in the reflexive space X.For a subsequence, we have u n ⇀ u weakly in X as n → +∞, which implies It follows by the property (S + ) and the continuity of Φ ′ that Moreover, since Φ ′ is an injection, we conclude that u = u.✷ Now we can give the proof of our main result.