Eigenvalues of the p ( x ) − biharmonic operator with indefinite weight under Neumann boundary conditions

abstract: In this paper we will study the existence of solutions for the nonhomogeneous elliptic equation with variable exponent ∆2 p(x) u = λV (x)|u|q(x)−2u, in a smooth bounded domain,under Neumann boundary conditions, where λ is a positive real number, p, q : Ω → R, are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.


Introduction
We are concerned here with the eigenvalue problem: where Ω is a bounded domain in R N with smooth boundary ∂Ω, N ≥ 1, ∆ 2 p(x) u = ∆(|∆u| p(x)−2 ∆u), is the p(x)-biharmonic operator, λ ≥ 0, p, q are continuous functions on Ω, and V is an indefinite weight function.The aim of this work is to study the existence of solutions for the nonhomogeneous eigenvalue problem (1.1), by considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.
In recent years, the study of differential equations and variational problems with p(x)-growth conditions is an interesting topic, which arises from nonlinear electrorheological fluids and other phenomena related to image processing, elasticity and the flow in porous media.In this context we refer to ( [10], [11], [6], [14], 196 S.Taarabti, Z.El Allali and K. Ben Hadddouch [12], [13]).This work is motivated by recent results in mathematical modeling of non Newtonian fluids and elastic mechanics, in particular, the electrorheological fluids (Smart fluids).This important class of fluids is characterized by change of viscosity, which is not easy to manipulate and depends on the electric field.These fluids, which are known under the name ER fluids, have many applications in electric mechanics, fluid dynamics etc...The same problem, for V (x) = 1 and p(x) = q(x) is studied by Ben Haddouch, El Allali, Ayoujil and Tsouli [2].The authors established the existence of a continuous family of eigenvalues by using the mountain pass lemma and Ekeland variational principle.
They proved the existence of a continuous family of eigenvalues by considering different situations concerning the growth rates involved in the above quoted problem.
In the case where p(x) = q(x), the authors in [14] investigated the eigenvalues of the p(x)−biharmonic with Navier boundary conditions.Ayoujil and El Amrouss [1], studied the same nonhomogeneous eigenvalue problem in the particular case when V (x) = 1.
In the case when max x∈Ω q(x) < min x∈Ω p(x) it can be proved that the energy functional associated to problem (1.2) when V (x) = 1, has a nontrivial minimum for any positive λ (see Theorem 3.1 in [1]).
In the case when min x∈Ω q(x) < min x∈Ω p(x) and q(x) has a subcritical growth, Ayoujil and El Amrouss [1] used the Ekelands variational principle in order to prove the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin.
In the case when max , by Theorem 3.8 in [1], for every λ > 0, the energy functional Φ λ corresponding to (1.2) has a mountain pass type critical point which is nontrivial and nonnegative, and hence Λ = (0, +∞).The authors established the existence of infinity many eigenvalues for problem (1.2) if q(x) = p(x) and V (x) = 1 by using an argument based on the Ljusternik-Schnirelman critical point theory.Denoting by Λ the set of all nonnegative eigenvalues, they showed that sup Λ = +∞.
Inspired by the above-mentioned paper, we will study the existence of solutions for the non-homogeneous elliptic eigenvalue problem Eigenvalues of the p(x)−biharmonic operator with indefinite weight 197

Preliminaries
In order to deal with p(x)− biharmonic operator problems, we need some results on spaces L p(x) (Ω) and W k,p(x) (Ω) and some properties of p(x)−biharmonic operator, which we will use later.Define the generalized Lebesgue space by: where p ∈ C + (Ω) and and for all x ∈ Ω and k ≥ 1 One introduces in L p(x) (Ω) the following norm and the space (L p(x) (Ω), |.| p(x) ) is a Banach.
Through this paper, we will consider the following space which is considered in ( [18]) and ( [2]).They have proved that X is a nonempty, well defined and closed subspace of W 2,p(x) (Ω).For this they have showed the following boundary trace embedding theorem for variable exponent Sobolev spaces. ) is a closed subspace of W 2,p(x) (Ω).
Proposition 2.5.If we put then for all u ∈ X then the following relations hold true In the case where u is a nontrivial solution, such a pair (u, λ) is called an eigenpair, λ is an eigenvalue and u is called an associated eigenfunction.
Proposition 2.6.If u ∈ X is a weak solution of (1.3) and u ∈ C 4 (Ω) then u is a classical solution of (1.3).
(2) there exists a continuous embedding of In what follows, we assume that the functions p, q ∈ C + (Ω).
Eigenvalues of the p(x)−biharmonic operator with indefinite weight 201

Main results and proofs
In this section we prove two theorems for problem (1.1).First, we prove the existence of a continuous family of eigenvalues for problem (1.1), in a neighborhood of the origin.
Then any λ > 0 is an eigenvalue for problem (1.1).Moreover, for any λ > 0 there exists a sequence (u n ) of nontrivial weak solutions for problem (1.1) such that u n → 0 in X.
In order to formulate the variational problem (1.1), let us introduce the functionals F, G, Φ λ : X −→ R defined by Denote by s ′ (x) the conjugate exponent of the function s(x) and put α(x) = s(x)q(x) s(x)−q(x) .Thus, by the proposition 2.7 the embeddings X ֒→ L s ′ (x)q(x) (Ω) and X ֒→ L α(x) (Ω) are compact and continuous.
The Euler-Lagrange functional associated with (1.1) is defined as Φ λ : X → R, We will show that Φ λ ∈ C 1 (X, R) and We only need to prove that G ∈ C 1 (X, R), that is, we show for all h ∈ X, and dG : X −→ X ′ is continous, where we denote by X ′ the dual space of X.For all h ∈ X, we have S.Taarabti, Z.El Allali and K. Ben Hadddouch The differentiation under the integral is allowed for t close to zero.indeed, for |t| < 1, using inequalities (2.2), (2.7) and condition H 1 (p, q, s), we have 1. Since X ֒→ L α(x) (Ω), X ֒→ L q(x) (Ω) and V ∈ L s(x) ((Ω)).On the other hand, we have X ֒→ L α(x) (Ω) (compact embedding).Furthermore, there exists c 1 such that |h| α(x) c 1 h .Therefore, by condition H 1 (p, q, s), we have for any h ∈ X.Thus there exists Eigenvalues of the p(x)−biharmonic operator with indefinite weight 203 Using the linearity of dG(u) and the above inequality we deduce that dG(u) ∈ X * .
The map defined in L q(x) (Ω) by u −→ |u| q(x)−2 u ∈ L q(x) q(x)−1 (Ω) is continuous.For the Fréchet differentiability, we conclude that G is Fréchet differentiable.Furthermore, for all u, v ∈ X.Similarly, we can also show that F ∈ C 1 (X, R).
Which implies that Φ λ ∈ C 1 (X, R) and for all u, v ∈ X.Thus the weak solutions of (1.1) coincide with the critical points of Φ λ .If such a weak solution exists and is nontrivial, then the corresponding λ is an eigenvalue of problem (1.1).Next, we write Φ ′ λ as We want to apply the symmetric mountain pass lemma in [8] to prove the Theorem 3.1.Then, I(u) admits a sequence of critical points u k such that where Γ k denote the family of closed symmetric subsets A of E such that 0 / ∈ A and γ(A) ≥ k with γ(A) is the genus of A, i.e., We start with two auxiliary results.
It remains to show that the functional Φ λ satisfies the (PS) condition to complete the proof.Let (u n ) ⊂ X be a (PS) sequence of Φ λ in X; that is, Then, by the coercivity of Φ λ , the sequence (u n ) is bounded in X.By the reflexivity of X, for a subsequence still denoted (u n ), we have Since q + < p − , it follows from proposition 2.7 that u n ⇀ u in L q(x) (Ω).We will show that In fact, from the Hölder type inequality, we have Since X is continuously embedded in L q(x) (Ω) and (u n ) is bounded in X, so u n is bounded in L q(x) (Ω).On the other hand, since the embedding X ֒→ L α(x) (Ω) is compact where α(x) = s(x)q(x) s(x)−q(x) , we deduce that |u n −u| α(x) −→ 0 as n −→ +∞.
we deduce that In view of (3.6) and (3.8), we obtain According to the fact that F ′ satisfies condition (S + ), we have u n → u in X.The proof is complete.Lemma 3.5.For each n ∈ N * , there exists an Now, we show that, for any n ∈ N * , there exist t n ∈]0, 1] such that sup u∈Hn(tn) Φ λ (u) < 0.
Indeed, for 0 < t ≤ 1, we have sup This completes the proof.
For applying Ekeland's variational principle.We start with two auxiliary results.

Theorem 3 . 3 .
(Symmetric mountain pass lemma) Let E be an infinite dimensional Banach space and I ∈ C 1 (E, R) satisfy the following two assumptions: (A1) I(u) is even, bounded from below, I(0) = 0 and I(u) satisfies the Palais-Smale condition (PS), namely, any sequence u n in E such that I(u n ) is bounded and I ′ (u n ) → 0 in E as n → ∞ has a convergent subsequence.(A2) For each k ∈ N, there exists an A k ∈ Γ k such that sup u∈A k I(u) < 0. 204 S.Taarabti, Z.El Allali and K. Ben Hadddouch applying Ekeland?s variational principle to the functionalΦ λ : B ρ (0) → R,Eigenvalues of the p(x)−biharmonic operator with indefinite weight 209 there exists u ε ∈ B ρ (0) such that Φ λ (u ε ) ≤ inf Bρ(0) n ), u n − u = 0.Using the last relation we deduce thatlim n→+∞ Ω |∆u n | p(x)−2 ∆u n ∆(u n − u)dx = 0.(3.17)From(3.17) and the fact that u n ⇀ u in X it follows that lim n→+∞ F ′ (u n ), u n − u = 0,