Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions

In this work we will study the eigenvalues for a fourth order elliptic equation with p(x)-growth conditions ∆2 p(x) u = λ|u|p(x)−2u, under Neumann boundary conditions, where p(x) is a continuous function defined on the bounded domain with p(x) > 1. Through the Ljusternik-Schnireleman theory on C1-manifold, we prove the existence of infinitely many eigenvalue sequences and supΛ = +∞, where Λ is the set of all 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 ( [8], [9], [10], [13], [11], [12]).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, 2000 Mathematics Subject Classification: 254 K. Ben Haddouch, Z.El Allali, N. Tsouli, S. El Habib and F. Kissi fluid dynamics etc...In the case where p(x) ≡ p (a constant), many authors have been interested in spectral problems including the p-Biharmonic operator (See [2], [3], [4], [5], [6], [7]), and in ( [1]), the authors have studied the problem (1.1), they have showed the existence of solution for the equation ∆ 2 p u = λm(x)|u| p−2 u under Neumann boundary conditions.In the variable exponent case, the authors in ( [13]) investigated the eigenvalues of the p(x)-biharmonic with Navier boundary conditions .In ( [14]), they considered the problem where p, q are continuous functions on Ω.Using the mountain pass lemma and Ekeland variational principle, they prove the existence of a continuous family of eigenvalues.
The main goal of this paper is to show the existence of solutions for the problem (1.1).We first prove the existence of positive eigenvalues of the following perturbed problem where ǫ is enough small (0 < ǫ < 1).
Through the Ljusternik-Schnireleman theory and by considering for each t > 0 the manifold where G(u) = Ω 1 p(x) |u| p(x) dx, we prove that for each t > 0 the problem (1.3) has a infinitely many eigenvalue sequences.And by tending ǫ −→ 0, we deduce that the problem (1.1) has infinitely many eigenvalue sequences.Our main results are stated in the following theorems: Theorem 1.1.For each t > 0 the problem (1.1) has infinitely many eigenpair sequences {(∓u n,t , ∓λ n,t )} such that λ n,t −→ ∞ as n −→ ∞.
Theorem 1.2.If there exist an open subset U ⊂ Ω and a point x 0 ∈ U such that p(x 0 ) < (or >)p(x) for all x ∈ ∂U , then λ * = 0

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: Existence of solutions for a fourth order eigenvalue problem 255 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 ) is a Banach.
Proposition 2.3.[15].For all p, r ∈ C + (Ω) such that r(x) ≤ p * k (x) for all x ∈ Ω, there is a continuous and compact embedding An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L p(x) (Ω) space, which is the mapping Through this paper, we will consider the following space which is considered by F.Moradi and all in ( [20]).They have proved that X is a nonempty, well defined and closed subspace of W 2,p(x) (Ω).Firstly they have showed the following boundary trace embedding theorem for variable exponent Sobolev spaces.
Theorem 2.5.[20] Let Ω be a bounded domain in IR N with C 2 boundary.If 2p(x) ≥ N ≥ 2 for all x ∈ Ω, then for all q ∈ C + (Ω) there is a continuous boundary trace embedding )

Proof
Consider the operator We prove that D is continuous from For this, we prove the continuity of the operator Using the second assertion of theorem 2.5 , we have u n −→ u in W 1,p(x) (∂Ω), which implies that ∇u n −→ ∇u in (L p(x) (∂Ω)) N , and then ∇ is continuous.Moreover, D = T • ∇ with T is the linear function defined as is the outer unit normal vector and On the other hand, we have which assert that T is continuous and then D is also continuous.Finally, since X = D −1 ({0}), it result that X is closed in W 2,p(x) (Ω).Hence, the proof of the proposition is completed.
A pair (u, λ) ∈ X × IR is a weak solution of (1.1) provided that In the case where u is nontrivial, such a pair (u, λ) is called an eigenpair, λ is an eigenvalue and u is called an associated eigenfunction.
Proposition 2.7.If u ∈ X is a weak solution of (1.1) and u ∈ C 4 (Ω) then u is a classical solution of (1.1).

Proof
Let u ∈ C 4 (Ω) be a weak solution of problem (1.1) then for every ϕ ∈ X, we have

8)
Existence of solutions for a fourth order eigenvalue problem
Definition 2.1.Let E be a real Banach space and A be a symmetric subset of E \ {0} witch is closed in E. We define the genus of A the number: and γ(A) = ∞ if does not exist such a map f .γ(∅) = 0 by definition.
Lemma 2.8.[18] Suppose that M is a closed symmetric C 1 -manifold of a real Banach space E and 0 / ∈ M .Suppose also that f ∈ C 1 (M, IR) is even and bounded bellow.Define where Γ j = {K ⊂ M : K is symmetric, compact and γ(K) ≥ j}.If Γ k = ∅ for some k ≥ 1 and if f satisfies (P S) c for all c = c j , j = 1, ..., k, then f has at least k distinct pairs of critical points.

Proof of main results
Let us consider a perturbation of problem (1.1) as follows where ǫ is enough small (0 < ǫ < 1).
Consider the functional Then According to the proposition 2.4 we have Proposition 3.1.For all u ∈ X, we have Theorem 3.2.For each t > 0, the problem (3.1) has infinitely many eigenpair sequences {(∓u n,t,ǫ , λ n,t,ǫ )} such that λ n,t,ǫ −→ ∞ as n −→ ∞ Proof Let us consider the functionals F ǫ , G : X −→ IR defined by It is well known that F ǫ , G ∈ C 1 (X, IR) and for all u, v ∈ X It is clear that (u, λ) is a weak solution of (3.1) if and only if We need the following result Proposition 3.3.(1) F ′ ǫ : X −→ X ′ is continuous, bounded and strictly monotone.

Proof
(1) Since F ′ ǫ is the Fréchet derivative of F ǫ , it follows that F ǫ is continuous and bounded.Let's define the sets ) for all (x, y) ∈ (IR N ) 2 , where x.y denotes the usual inner product in IR N , we obtain for all u, v ∈ X such that u = v which implies that F ′ ǫ is strictly monotone.
(2) We consider (u n ) n a sequence of X such that u n ⇀ u in X and lim sup n→+∞ 262 K. Ben Haddouch, Z.El Allali, N. Tsouli, S. El Habib and F. Kissi From the proposition 3.1, it suffices to show that by the monotonicity of F ′ ǫ , we have and since u n ⇀ u in X, we deduce that We consider By the compact embedding of X into L p(x) (Ω), it follows that and the Young's inequality yields that From (3.5) and since ϕ n ≥ 0, we can consider that 0 > 1, and the fact that 2 p(x) < 2, the inequality (3.7) becomes Note that,

Similarly, we have
Vp We conclude that (3) We prove now that F ′ ǫ is an homeomorphism.First, by the strict monotonicity, F ′ ǫ is an injection.Furthermore, for any u ∈ X with u > 1, we have i.e.F ′ ǫ is coercive.Thus, F ′ ǫ is a surjection in view of Minty-Browder theorem (see theorem 26.A(d) in [19]).Hence, F ′ ǫ has an inverse mapping (F ′ ǫ ) −1 : X ′ −→ X.Therefore, the continuity of (F ′ ǫ ) −1 is sufficient to ensure F ′ ǫ to be an homeomorphism.
by coercivity of F ′ ǫ , one deducts that the sequence (u n ) is bounded in the reflexive space X.For a subsequence (u n ), we have u n ⇀ u in X, which implies that It follows by the second assertion and the continuity of ǫ is an injection, we conclude that u = u Proposition 3.4.G ′ : X −→ X ′ is sequentially weakly-strongly continuous, namely, For any v ∈ X, by Hölder's inequality in X and continuous embedding of X in to L p(x) (Ω), it's follows that By using the compact embedding of X in to L p(x) (Ω), we have To solve the eigenvalue problem (3.2), we will use the Ljusternik-schnirelmann theory on C 1 -manifolds (see [18] corollary 4.1).For any t > 0, denote by We know that for all x ∈ Ω, we have Proposition 3.5.F ǫ satisfies the (P S) condition, namely, any sequence Hence, v / ∈ T u (M t ); therefore We consider P : X −→ T u (M t ) the natural projection.Then, for every w ∈ X, there exists a unique β ∈ IR such that w = P w + βv.

267
..., u n }.Then, dim(E n ) = n.Note that the map It is clear that f (0, u) = 0 and f (s, u) is nondecreasing with respect to s. More, for s > 1 we have f (s, u) ≥ s p − G(u), and so lim s−→+∞ f (s, u) = +∞.Therefore, for every u ∈ S n fixed, there is a unique value s = s(u) > 0 such that f (s(u), u) = 1.
On the other hand, since The implicit theorem implies that the map u −→ s(u) is continuous and even by uniqueness.Now we take the compact K n = M t ∩ E n .Since the map h : S n −→ K n defined by h(u) = s(u).u is continuous and odd, it follows by the property of genus that γ(K n ) ≥ n.This completes the proof.
Proof X is a reflexive and separable space, there are {e i } ⊂ X and {f i } ⊂ X ′ such that f i , e i = δ i,j (Kronecker symbol).We have X = span{e i : i ∈ IN * } and X ′ = span w * {f i : i ∈ IN * }.
For n = 1, 2, ..., denote by Using the following Proposition 3.8.[21] Assume that ϕ : X −→ IR is weakly-strongly continuous and ϕ(0) = 0, r > 0 is a given positive number.Then Indeed, by contradiction, assume that there exist c 0 > 0 and {u n } ⊂ M t ∩ Z n such that u n ≤ c 0 for all n ∈ IN.Therefore, That is a contradiction with the proposition 3.8.From (3.9), for each c > 1; there exists n 0 ∈ IN such that for all n > n 0 and u ∈ M t ∩ Z n , u > c.On the other hand, for any K ⊂ M t , compact and symmetric, we have γ(K ∩ Y n−1 ) ≤ n − 1.
As cod(Z n ) ≤ n−1 and by the property of genus, for K ⊂ Σ n,t , we have K ∩Z n = ∅.This achieves the proof.Applying proposition 3.5, lemma 3.6 and Ljusternik-schnireleman theory to the problem 3.1, we have for each n ∈ IN, c n,t,ǫ is a critical value of F ǫ on submanifold M t , such that 0 < c n,t,ǫ ≤ c n+1,t,ǫ , c n,t,ǫ −→ +∞ as n −→ +∞.