Multiplicity Results for a Fourth Order Quasi-Linear Problems

(P) { ∆pu = f(x, u) in Ω u = ∆u = 0 on ∂Ω, where Ω is a bounded domain in R with smooth boundary ∂Ω, N ≥ 1, ∆pu = ∆(|∆u|∆u), is the p-biharmonic operator, 1 < p < ∞ and f : Ω× R → R is a Carathéodory function satisfying the subcritical growth condition: (F0) |f(x, t)| ≤ c(1 + |t| ), ∀t ∈ R, a.e. x ∈ Ω, for some c > 0, and 1 ≤ q < p where p = Np N−2p if 1 < 2p < N and p ∗ = +∞ if N ≤ 2p. Observe that, if f(x, 0) ≡ 0, then the problem (P) has a trivial solution u ≡ 0. We are interested in finding multiple nontrivial solutions of (P) in the Sobolev space W (Ω) ∩W 1,p 0 (Ω), equipped with the norm

Observe that, if f (x, 0) ≡ 0, then the problem (P) has a trivial solution u ≡ 0. We are interested in finding multiple nontrivial solutions of (P) in the Sobolev space W 2,p (Ω) ∩ W 1,p 0 (Ω), equipped with the norm u = ( It is well known that the functional Φ : W 2,p (Ω) ∩ W  for every ϕ ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω).Moreover, the critical points of Φ are weak solutions for (P).Notice that for the eigenvalue problem as for the p-Laplacian eigenvalue problem with Dirichlet boundary data, is the sequence of eigenvalues, where A n = {K ⊂ N : K is compact, symmetric and γ(K) ≥ n} and Here γ(K) indicate the genus of K.It has been recently proved by P. Drábek and M. Ôtani [4] that (1.1) has the least eigenvalue which is simple, positive and has an associated normalized eigenfunction ϕ 1 which is positive in Ω.It is also known, (see [4]), that there exists δ > 0 such that (λ 1 (p), λ 1 (p) + δ) that not contain other eigenvalues.
The existence of solutions of p-biharmonic equation has been studied by several authors see [1,4,9,11] and the reference therein.It will be seen that critical groups and Morse Theory, developed by Chang [3] or Mawhin and Willem [10], are the main tools used to solve our problem.The main point in this theory is to introduce the critical groups of an isolated critical point.With this aim, we need to suppose a conditions that give us information about the behavior of the perturbed function f (x, t) or its primitive F (x, t) = t 0 f (x, s) ds near infinity and near zero.More precisely, the following conditions are assumed: The main result reads as follows.
The second purpose of this paper is to show the existence of at least two nontrivial solutions of problem (P) under the following assumptions: we can state the following result.Theorem 1.2.Under (F 0 ), (F 6 ) and (F 7 ), the problem (P) has at least two nontrivial solutions.
Remark 1.2.In Theorem 1.4 [9], the authors established the existence of at least two nontrivial solutions of problem (P), under (F 0 ), (F 6 ) and the following hypothesis.
Note that our condition (F 7 ) is weaker than (F ′ 7 ).For finding critical points of Φ, by applying minimax methods, we will use the following compactness condition, introduced by Cerami [2], which is a generalization of the classical Palais-Smale type (PS).
The paper is organized as follows.In section 2 we introduce some auxiliary results.In Section 3, we will prove the existence of at least nontrivial solution by combining the minimax method and Morse theory.In section 4, we will give the proof of Theorem 1.2.

Critical Groups
In this section, we investigate the critical groups at zero and at a mountain pass type.To proceed, some concepts are needed.Let X denote the generalized Sobolev space Denote by H q (A, B) the q-th homology group of the topological pair (A, B) with integer coefficient.The critical groups of Φ at an isolated critical point u ∈ K c are defined by where U is a closed neighborhood of u.Moreover, it is known that C q (Φ, u) is independent of the choice of U due to the excision property of homology.We refer the readers to [3,10] for more information.
Recall that in the case when Φ satisfies the Cerami condition and for is finite, we have the following Morse relations between the Morse critical groups and homological characterization of subset sets: (2.1) Now, we will show that the critical groups of Φ at zero are trivial.
Proof.Let B ρ = {u ∈ X, u ≤ ρ}, ρ > 0 which is to be chosen later.The idea of the proof is to construct a retraction of B ρ \ {0} to B ρ ∩ Φ 0 \ {0} and to prove that B ρ ∩ Φ 0 is contractible in itself.For this purpose, we need to analyze the local properties of Φ near zero.Thus, some technical affirmations must be proved.Claim 1.Under (F 0 ) and (F 5 ), zero is local maximum for the functional Φ(su), s ∈ R, for u = 0.
In fact, it follows from the first condition of (F 5 ), there exists a constant c 0 > 0 such that Using (F 0 ) and (2.2), we get for some q ∈ (p, p * ) and c 1 > 0.
Thus, using (2.6) and (2.9) and the Implicit Function Theorem we get that the mapping T is continuous.
Next, we define a mapping η : Since T (u) = 1 as Φ(u) = 0, the continuity of η follows from the continuity of T .Obviously, By the fact that retracts of contractible space are also contractible, From the homology exact sequence, one has The proof of lemma 2.1 is completed.✷ We will use the following lemma, which is proved with (PS) condition see for example [10].

If Φ satisfies the (C) condition over X and if each critical point of
where ∂ is the boundary homomorphism and i * is induced by the inclusion mapping i : (Φ c−ε , ∅) → (Φ c+ε , ∅).The definition of c implies that u 0 and u 1 are path connected in Φ c+ε but not in Φ c−ε .Thus, ker i * = {0} [3,10] and, by exactness, The proof is based on the following minimax theorem due to the first author [5, Theorem 3.5], with Cerami condition.
Proof.(i) First, we verify that the Palais-Small condition is satisfied on the bounded subsets of X.Let (u n ) ⊂ X be bounded such that Passing if necessary to a subsequence, we may assume that From (3.1) and (3.2), we have By the Hölder inequality, we obtain Thus, it follows from (3.3) and (3.4) that ∆ 2 p u n , u n − u → 0. Since , ∆ 2 p is of type S + ( see [4] ), we deduce that u n → u strongly in X.
Now, we will show that (ii) is satisfied for every c ∈ R. By contradiction, let Therefore, Taking v n = un un , clearly v n is bounded in X.So, there is a function v ∈ X and a subsequence still denote by (v n ) such that On the other hand, in view (F 0 ) and (F 3 ), it follows that Combining relations (3.5) and (3.8), we obtain Dividing by u n and passing to the limit, we conclude and consequently v = 0.
Let Hence, Using (3.9) and Fatou's lemma, one deduce This contradicts (3.6).✷ Next, we will prove the geometric conditions of Theorem 3.1.Let denote E(λ 1 ) the eigenspace associated to the eigenvalue λ 1 Lemma 3.2.Under the hypothesis (F 0 ), (F 3 ) and (F 4 ), we have: Proof.(i) For every v ∈ E(λ 1 ), there exists t ∈ R such that v = tϕ 1 .Therefore, using (F 4 ), we write (ii) By the Ljusternik-Schnirelmann theory, we write Then, for all K ∈ A 2 , and all ε > 0, there exists v K ∈ K such that Indeed, if 0 ∈ K, we take v K = 0. Otherwise, we consider the odd mapping By the genus properties, we have γ(g(K)) ≥ 2, and by the definition of λ 2 , there exist On the other hand, the two assumptions (F 0 ) and (F 3 ) implies for some constant C > 0. Consequently, one deduce from (3.10) and (3.11) that The argument is similar for In this section we prove Theorem 1.2 via the following abstract critical point theorem.
Theorem 4.1 ( [8]).Let X be a real Banach pace and let Φ ∈ C 1 (X, R) be bounded from below an satisfying the Palais-Smale condition.Assume that Φ has a critical point u which is homologically nontrivial, that is, C j (Φ, u) = {0} for some j, and it is not a minimizer for Φ.Then Φ admits at least three critical points.
In order to apply Theorem 4.1, we need the following lemmas.First, we recall the notion of "Local Linking", which was initially introduced by Liu and Li [7].Definition 4.1.Let X be a real Banach space such that X = V ⊕ W , where V and W are closed subspace of X.Let Φ : X → R be a C 1 -functional.We say that Φ has a local linking near the origin 0 (with respect to the decomposition X = V ⊕ W ), if there exists ρ > 0 such that u ∈ V : u ≤ ρ =⇒ Φ(u) ≤ 0, u ∈ W : 0 < u ≤ ρ =⇒ Φ(u) > 0. (4.1)