Abstract

The existence of solution for a fourth-order nonlinear partial differential equation (PDE) class involving p-biharmonic operator , is proved by applying mountain pass theorem and a local minimization.

1. Introduction and Assumptions

The goal of this paper is to investigate the existence of nontrivial solutions for a class of nonlinear partial differential equations of the fourth order of the form where is a smooth bounded domain in , is a parameter which plays the role of eigenvalue, and and are real numbers such that and where is the critical Sobolev exponent defined by if and if , and is an indefinite weight in with being so that and (the Lebesgue measure) is an operator of the fourth order called the -biharmonic operator. For , the linear operator is the iterated Laplace that multiplied with positive constant occurs often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator denoted is the celebrated Green operator [1]. The boundary condition in is of the compatibility type when which can be considered as the following Hamiltonian system of the two coupled equations: with , and is the Hölder conjugate exponent of , by a transformation of a problem to a known Poisson's problem and using the properties of the operator-solution stated by Agmon-Douglis-Nirenberg; see [2]. Notice that a similar system as was considered recently by [3] in the restrictive case . Our approach it quite different. Note that, in the case , is -homogeneous, so we are dealing a quasilinear eigenvalue problem which is considered differently. This is few considered by [4] in the particular case and is smooth; and by [5, 6] with as a boundary condition, for any bounded domain.  

Note also that the nonhomogeneous case is not considered there. Here we seek nontrivial solutions for by distinguishing between two subcritical cases and which means that the critical points of the associated energy functionalare defined in . One solution is obtained by applying classical Mountain Pass Theorem and the other solution by a local minimization technique. The restrictive case and is substituted by the well-known -Laplacian was studied by Azorero and Alonso [7] and Elkhalil and Touzani [8] by using the fundamental properties of the first eigenvalue of the Dirichlet -Laplacian problem.

It is important to indicate here that condition (1.2) is optimal to ensure the Palais-Smale (PS) condition is satisfied. Notice that our results are investigated without any condition on the parameter related to the spectrum of (when ). On the other hand the condition (1.3) is assumed to have eventually nontrivial solutions.

The rest of this paper is divided in two sections as follows. In Section 2, we introduce some preliminary results and we give some technical lemmas, and in Section 3, we establish our main results.

2. Preliminaries

First, we introduce some preliminary results that we will need and some lemmas that are the key point of our results. We solve the problem in the space endowed with the norm equipped with this norm is an uniformly convex Banach space for . Hereafter, is the -norm, will denote the duality between and its dual .

Definition 2.1. is a weak solution of if, and only if, for all we have

That is, is a critical point of the energy functional associated to defined on by

Lemma 2.2. For any verifying (1.2) and satisfying (1.1) there exists a constant such that and the map is strongly continuous from into .

Proof. To establish (2.4), we will divide the proof to three steps with respect to exponents and .Step 1 (). Fixing and using Hölder's inequality we have where Such exponent exists. Indeed: if , we obtain Thus, it suffices to take so that If , we have . So, Hence, Therefore, it suffices to take such that Step 2 (). In this case, for any , . Thus, for (some exists because ), we get Hölder's inequality yields Step 3 (). In this case we have and . Thus, To prove the continuity of , we argue as follows.
Let such that in . Thus, in , that is, We prove it in the case , and with satisfying (i).
Using (2.6) or (2.10), we obtain by calculation Therefore, the desired result can be obtained since the limit holds by using the continuity of the Nemytskii operator from and the right embedding of Sobolev space.

Remark 2.3. If is a solution of associated to , then, for any , is also a weak solution of associated to . Hence, we can reduce the problem to .

Lemma 2.4. For we have the following assertions (i)if then is bounded from below, (ii)if thus, there exist two reel and such that: if , if .

Proof. Step 1 (). We have for a positive constant .
If , then
If , thenStep 2 (). From (2.18) we deduce that If , then there exists such that If , we get The estimation above completes the proof.

Proposition 2.5. If the pair satisfies (1.1) with , then satisfies the (PS) condition.

Proof. Let be a sequence of Palais-Smale of in . Thus there exists such that and for any there is such that Claim that is a bounded sequence in .Step 1 (). By (2.18) we deduce that is coercive, then the claim follows.Step 2 (). Equation implies that, for all , Thanks to , we obtain, Multiplying (2.24) by and adding with (2.18), we obtain, Hence, there exists a positive constant such that So is bounded in the two cases in and the claim is concluded. Consequently, there exists a subsequence still denoted converges weakly for some and strongly in and in for all satisfies (1.2).
Let Thus Since is strongly continuous from into , we deduce from , as in (2.29), that On the other hand, we have This and (2.30) imply that Finally, since is uniformly convex, the proof is achieved.

3. Main Results

3.1. The Case : Local Minimization Technique

In this subsection, we show that the problem has a sequence of weak solution by using the results of Lusternik-Shnirleman [9]. In other words, we use a local minimization for the corresponding energy functional.

Let and set We will show that the sequence , defined by (3.1) is critical values of . Here and in the following is the genus of , that is, the smallest integer such that there is an odd continuous map from into .

Lemma 3.1. For any ,

Proof. is separable. Therefore, for any , there exists a finite sequence of functions in linearly independent such that for and thanks to (1.3) we can choose such that . Let be a subspace in spend by of dimension .
If , then, there exist real numbers such that .
Thus, . Hence, the map is a norm in . Consequently, there exists a constant such that Set with . It is clear that is a closed neighborhood, symmetric, compact not containing 0. Finally, by the property of genus we get and .

Theorem 3.2. Let be a sequence defined by (3.1). Then, we have the following statements: (i)For any , there exist distinct pairs critical points of the functional . (ii)If then where

Proof. From the result of [10], it suffices to prove .
Recall that the functional is even, bounded from below and .
In view of Lemma 2.4, is of class on ; satisfying the (PS) condition.
Hence, these properties confirm the hypotheses of Clark's Lemma cf. [11] which proves .

Remark 3.3. The set is compact. Indeed, let be a sequence in , that is, It is clear that, for , Since is bounded from below, by Lemma 2.4, is a sequence of Palais-Smale (PS). Thus, possesses a convergent subsequence.
Consequently, is compact in and the functional has an infinity of critical points.

3.2. The Case : Mountain Pass Theorem

Here, we prove the existence of solutions of problem , by using Mountain Pass theorem [11].

Lemma 3.4. Suppose that and with satisfying (1.2). Hence there exists such that

Proof. We have for any and , From (1.3), there exists such that . Thus, Hence, there is large enough so that . To achieve the proof, it suffices to take .

Consequently, we conclude the following statements. (a) is of class , even and . (b) satisfies the (PS) condition.(c)There exist two positive reel , such that (d)There exists such that .

Now, we can establish the following theorem.

Theorem 3.5. If , then the value defined by is a critical value of .
Here,

Proof. In view of Lemma 2.4., Proposition 2.5, and properties (a)–(d), is a critical value of by applying the Mountain Pass Theorem due to [11].

Acknowledgment

The author gratefully acknowledges the financial support provided by Al-Imam Muhammed Ibn Saud Islamic University during this research.