Existence and multiplicity of solutions for class of Navier boundary p-biharmonic problem near resonance

where Ω ⊂ R (N ≥ 1) is a bounded smooth domain, p > 1, ρ ∈ C(Ω) with infΩ ρ(x) > 0, f : Ω × R → R is a Carathéodory function and m ∈ C(Ω) is nonnegative weight functions. The investigation of existence and multiplicity of solutions for problems involving p-biharmonic operator has drawn the attention of many authors, see reference. In [4], Li and Tang considered the following Navier boundary value problem

1 Introduction and main results 83 2 Preliminaries and proofs of Theorems 86

Introduction and main results
In this article, we are concerned with the following elliptic problem of p-biharmonic type ∆(ρ(x)|∆u| p−2 ∆u) = λm(x)|u| p−2 u + f (x, u) + h(x) in Ω, u = ∆u = 0 on ∂Ω. (1.1) where Ω ⊂ R N (N ≥ 1) is a bounded smooth domain, p > 1, ρ ∈ C(Ω) with inf Ω ρ(x) > 0, f : Ω × R → R is a Carathéodory function and m ∈ C(Ω) is nonnegative weight functions.The investigation of existence and multiplicity of solutions for problems involving p-biharmonic operator has drawn the attention of many authors, see reference.
In [4], Li and Tang considered the following Navier boundary value problem where p > max 1, N 2 and λ, µ ≥ 0. Under suitable assumptions the existence of at least three weak solutions is established.In [6], Ma and Pelicer study a multiplicity for the perturbed p-Laplacian equation where λ is near λ 1 , the principal eigenvalue of the weighted problem they proved the existence of one or three solutions.
In the present paper, we study problem (1.1) that result was extended to the p-biharmonic operator in bounded domains, with the weight functions.We were inspired by Ma and Pelicer [6] in which problems involving the p-laplacian operator is studied.Our technical approach is based on Ekeland's variational principle, Mountain pass theorem and saddle point theorem.We assume that f satisfies the following conditions (F 1 ) There exists a real a > 0 and a function b We introduce the space X := W 2,p (Ω) ∩ W 1,p 0 (Ω), which is a reflexive Banach space endowed with the norm , (see, e.g., [1,10]).
Definition 1.1.We say that u ∈ X is a weak solution of problem (1.1) if The corresponding energy functional of problem (1.1) is given by it is well known that I ∈ C 1 (X, R), with derivative at point u ∈ X is given by for every ϕ ∈ X.Consequently, the critical points of the functional I correspond to the weak solutions of the problem (1.1).
Theorem 1.2.let X be a real Banach space and I : X → R be a C 1 functional satisfying the Palais-Smale condition.Furthermore assume that I(0) = 0 and that the following conditions hold: (i) there exits a number r > 0 such that I| ∂Br ≥ 0 (ii) there is an element e ∈ X\B r with I(e) ≤ 0. Then the real number c, characterized as

I(γ(t))
where is a critical value of I with c ≥ 0. If c = 0, there exists a critical point of I on ∂B r corresponding to the critical value 0.
Theorem 1.3.Let X be a Banach space.Let I : X → R be a C 1 functional that satisfies the Palais-Smale condition, and suppose that X = V ⊕ W, with V a finite dimensional subspace of X.If there exists R > 0 such that max v∈V,||v||=R then I has a least a critical point on X.
Now we are ready to state our main result.
Theorem 1.4.Assume that (F 1 ) holds.If in addition ) problem (1.1) has at least three solutions when λ is sufficiently close to λ 1 from left.

Preliminaries and proofs of Theorems
Let denote V = ϕ 1 the linear spans of ϕ 1 and Then we can decompose X as a direct sum of V and W. In fact, let u ∈ X, writing where w ∈ X, and We begin by establishing the existence of λ for which (1.7) holds.
Existence and multiplicity of solutions 87 This value is attained in W. To see why this is so, let is bounded in X and therefore, up to subsequence, we may assume that u n ⇀ u weakly in X and u n → u strongly in L p (Ω).
From the strong convergence of the sequence in L p (Ω) we obtain and Proof: From (F 1), we have By Hölder's and Sobolev's inequalities, it follows from (1.5) that where C 1 , C 2 and C 3 are the embedding constants of Sobolev.Since λ < λ 1 and σ < p, I is coercive.Similarly, let u ∈ W, by Lemma 2.1, for λ < λ 1 , we have Hence I is bounded from below on W.Moreover, we can find a constant m independent of λ such that inf W I(u) ≥ m. ✷ Lemma 2.3.Assume that (F 1) and (1.9) hold.Then, for λ < λ 1 sufficiently close to λ 1 , there exist t − < 0 < t + such that where m is given by Lemma 2.2.
Proof: By definition of λ 1 and (1.10), we have From (1.9), for t > 0 large enough, we have Existence and multiplicity of solutions 89 by Fatou's Lemma, we get so, there exists t + > 0 such that For λ 1 − pλ1 (t + ) p < λ < λ 1 , (2.5) and (2.6) imply Similarly, we get I(t − ϕ 1 ) < m, for some t − < 0. ✷ Proof: (Theorem 1.4) First we show that I satisfies the (P S) condition in X, that is for every sequence such that possesses a convergent subsequence.Let (u n ) ⊂ X be a (P S) sequence.Since I is coercive, (u n ) is bounded in X, so up to subsequence, we may assume that u n ⇀ u weakly in X. Therefore (2.8) By Hölder's inequality, we have (2.10) , and hence there exists Since the right side of the last inequality belongs to L 1 (Ω), it follows from Lebesgue theorem that By using the fact that u n ⇀ u in L p ⋆ (Ω), we deduce that Combining (2.8), (2.10) and (2.11) we obtain In the same way, we obtain Therefore, the Hölder inequality imply that By the uniform convexity of X, it follows that u n → u strongly in X and I satisfies the (P S) condition.Next, let which is impossible.Therefore u ∈ Λ + , and hence I satisfies the (P S) c,Λ + for all c < m.Similarly, I satisfies the (P S) c,Λ − for all c < m.
In view of Lemma 2.3 for λ < λ 1 sufficiently close to λ 1 , we have where Since J also satisfies the (P S) condition and J ′ = I ′ , it follows from the Mountain pass theorem 1.2 that c is a critical value of I. Noting that all paths joining u − to u + pass through W, so c ≥ m.Therefore the third solution is obtained, and the proof of theorem is complete.✷ Proof: (Theorem 1.5)The proof will be divided in some steps.
Step 1 (the growth of F ).We prove that for some (2.15) From (F 2 ), we have Noting that F (x,u) |u| p → 0 as u → ∞, thus after integration from u > 0 to +∞, we see that and (2.15) follows.
Step 2 (the Palais-Smale condition).Let (u n ) be a sequence satisfying (2.7), we note that (2.16) From the boundedness of I ′ (u n ), u n − pI(u n ), we deduce that (u n ) is bounded in X.By a similar argument as in the proof of Theorem 1.4, we conclude that (u n ) possesses a convergent subsequence in X.
Step 3 (the saddle point theorem).Using again Lemma 2.1, we get Since λ < λ, inf w∈W I(w) > −∞. (2.17) On the other hand, by (2.15) we see that It follows from λ ≥ λ 1 and 1 < µ < p that Hence, I satisfies the hypotheses of Theorem 1.3, and there exists a critical point of I, that is a solution of (1.1).✷