Existence of solutions for a boundary problem involving p ( x )-biharmonic operator

In this paper, we establish the existence of at least three solutions to a boundary problem involving the p(x)-biharmonic operator. Our technical approach is based on theorem obtained by B. Ricceri’s variational principle and local mountain pass theorem without (Palais-Smale) condition.


Introduction
The study of various mathematical problems with variable exponent have received a lot of attention in recent years [1,14].Fourth order equations appears in many contexts.Some of these problems come from different areas of applied mathematics and physics such as Micro Electro-Mechanical systems, surface diffusion on solids, flow in Hele-Shaw cells (see [9]).In addition, this type of equation can describe the static from change of beam or the sport of rigid body, there are many authors have pointed out that type of non linearity furnishes a model to study traveling waves in suspension bridges (see [5,11]).
In this paper, we consider the following p(x)−biharmonic problem with a boundary condition, Denote by F (x, t) = t 0 f (x, s)ds , G(x, t) = t 0 g(x, s)ds , p − := inf x∈Ω p(x) and p + := sup x∈Ω p(x).Throughout this paper, we suppose the following assumptions:

(F ′
2 ) There exist ξ ∈ R such that In the case B = B 1 and T = T 1 , we claim the following theorem.
The case B = B 2 , T = T 2 , we have the following result.
Many authors consider the existence of multiple nontrivial solutions for some fourth order problems [11,16].In particular, Li and Tang [10] consider the pbiharmonic equation.Using the modified three critical points theorem of B. Ricceri they get at least three solutions.The p(x)−biharmonic operator possesses more complicated nonlinearities than p−biharmonic, for example, it is inhomogeneous.Recently, in [4] A. Ayoujil and A. R. El Amrouss interested to the spectrum of a fourth order elliptic equation with variable exponent.They proved the existence of infinitely many eigenvalue sequences and supΛ = +∞, where Λ is the set of all eigenvalues.Moreover, they present some sufficient conditions for inf Λ = 0.
The technical approach is based on the Ricceri's variational principle and local mountain pass theorem [3], without Palais-Smale condition.One of the first result in this direction was obtained by Shao-Gao Deng [6] for the p(x) − laplacien, here, we borrow some ideas from that work.The purpose of this work is to improve the results of [6] and extend them to the case of p(x)-biharmonic equation with Navier and Neumann condition.
This article consists of three sections.In section 2, we start with some preliminary basic results on theory of Lesbegue-Sobolev spaces with variables exponent (we refer to the book of Musielak [13] , Mihȃilescu and Rȃdulescu [12]), we recall Ricceri's variational principle with some results which are needed later.In section 3, we give the proof of the main result.

Variable exponent space and Sobolev Spaces
In order to deal with the problem (P), we need some theory of variable exponent Sobolev Space.For convenience, we only recall some basic facts which will be used later.Suppose that Ω ⊂ R N be a bounded domain with smooth boundary ∂Ω.
becomes a Banach space separable and reflexive space.
The proof is similar to proof in [7] , We denote by We define , The following proposition will be used later, (2) the mapping L ′ : X → X ′ is a strictly monotone, bounded homeomorphism and is of type S + , namely, u n ⇀ u and lim sup n→∞ L ′ (u n )(u n − u) ≤ 0 implies that u n → u, where → and ⇀ denote the strong and weak convergence respectively.
By the continuity and convexity of L, we deduce that L is sequentially weakly lower semi continuous .
(2) Since L ′ is Fréchet derivative of L then L is continuous and bounded.We set By the elementary inequalities, we have ∀x, where x.y denotes the usual inner product in R N .Then for all u, v ∈ X such that by the monotonicity of L ′ , we claim that By the compact embedding of X into L p(x) (Ω), where 1 p(x) + 1 q(x) = 1 for all x ∈ Ω.It follows that, (2.4) Thanks to the above inequalities,

It results that
Up

6)
Existence of solutions 185 From (2.4) and since χ n ≥ 0, one consider that 2 > 1 and the fact that 2 p(x) < 2, inequality (2.6) becomes Note that, Ω δ p(x) n dx is bounded, which implies Similarly we can have Hence, it result that Finally, (2.1) is given by combining (2.5) and (2.7).It remains to show that L ′ is is a homeomorphism.In view of strict monotonicity of L ′ which implies the injectivity of L ′ .Moreover, L ′ is a coercive.Indeed, since p −1 > 1, for each u ∈ X such that u ≥ 1 we have Consequently, thanks to a Minty-Browder [15], L ′ is surjective and admits an inverse mapping.It suffices to show the continuity of By the coercivity of L ′ , one deducts that the sequence (u n ) is bounded in the reflexive space X.For a subsequence, we have u n ⇀ u in X, which implies Since L ′ is of (S + ) type and continuous, it follows that Moreover, since L ′ is an injection, we deduce that u = u. 2 Proposition 2.6.let σ(u) = Ω G(x, u)dx, then σ is a C 1 in L q(x) (Ω) and σ ′ are weakly-strongly continuous, i.e u n ⇀ u implies σ ′ (u n ) → σ ′ (u).
step (1): To show that v 0 = 0 is strictly local minimizer of I, we follow the same procedure as in step (1) in the previous proof.

the weak topology. Definition 2 . 3 .Definition 2 . 4 .Theorem 2 . 1 .
Let D a bounded open subset of X and c < b is called Ricceri box of I with the type (c, d) if Let Y be a Banach space, G 0 and G be two bounded open subset of Y with G 0 ⊂ G and φ: Y → R a functional.(G 0 , G) is a valley box of φ if sup G0 (see[6,8]) Assume that I, J : X → R are sequentially weakly lower semi continuous and G is a Ricceri block of I with type ρ.Letλ * = sup x∈G ρ − I(x) J(x) − inf G W J then for each λ ∈]0, λ * [ , the restriction of I + λJ to G W achieves its infimum at some x * ∈ G, so x * is a local minimizer of I + λJ.Remark 2.2.i) let u * ∈ X astrictly local minimizer of I, then for ǫ > 0 small enough, we have inf ∂B(u,ǫ) I > I(u * ) i.e B(u * , ǫ) is a Recceri box of I. ii) In fact, by proposition 2.6) in [8], I, J : X → R are sequentially weakly lower semi continuous.Proposition 2.7.[8] Suppose that G is a Ricceri box of I with type (c, b) and I : X → R continuous.Then for every ρ ∈]c, b] we have I −1 (] − ∞, ρ[) ∩ G is a Ricceri block of I with type ρ.By Proposition 2.5, Remark 2.2 and Theorem 2.1 we obtain the following result, Proposition 2.8.[6] Suppose that I, J : X → R are continuous.For some r > 0, u 1 ∈ B(u 0 , r), I(u 0 ) = inf B(u0,r) I = c 0 ; inf ∂B(x0,r) I = b > c 0 and u 1 is a strictly local minimizer of I and I(u 1 ) = c 1 > c 0 .Then for ǫ > 0 small enough and ρ 1 > c 1 , ρ 0 ∈]c 0 , min{b, c 1 }[ and ∀λ ∈]0, λ * [, I + λJ has at least two local minima u * 0 , u * 1 in B(u 0 , r).Where u * 0