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Weak Solutions for a Class of (p,q) Laplacian quasilinear Elliptic System with Different Weights

Tahar Bouali*, Rafik Guefaifiai

Department of Mathematics, University Tebessa, Algeria

Abstract: Using variational methods, we study the existence of weak solutions for the degenerate quasilinear elliptic system where is a smooth bounded domain, stands for the gradient of -function the weights are allowed to vanish somewhere, the primitive is intimately related to the first eigenvalue of a corresponding quasilinear system

Keywords: quasilinear elliptic system, Palais-Smale condition, mountain pass theorem, existence

Received March 30, 2014; Revised November 09, 2014; Accepted November 16, 2014

Citation: Tahar Bouali, Rafik Guefaifiai, Weak Solutions for a Class of (p,q) Laplacian quasilinear Elliptic System with Different Weights. International Journal of Partial Differential Equations and Applications, 2 (5). 86-90. January 2014. doi:10.12691/ijpdea-2-5-1

1. Introduction

In this paper, we are concerned with the quasilinear elliptic system

[picture] | (1.1)

where is a smooth bounded domain in , nonnegative real number, stands for the gradient of in the variable . We point out that in the case, problem (1:1) has been studied in many papers. For more details about this kind of systems, we refer to [4,8,9,11,12,13,15,19], in which the authors used various methods to get the existence of solutions. The degeneracy of this system is considered in the sense that the measurable, non-negative diffusion coefficients , are allowed to vanish in (as well as at the boundary ) and/or to blow up in . The point of departure for the consideration of suitable assumptions on the diffusion coefficients

Let us introduce the function space which consists of functions such that and for some satisfying Then for the weight functions we assume the following hypothesis:

There exist functions in the space for some and in the space , for some, such that

[picture] | (1.2)

a.e. in for some constants

We consider the weighted Sobolev spaces and to be defined as the closures of with respect to the norms

and set It is clear that is a reflexive Banach space under the norm

For more details about the space setting we refer to [10] and the references therein. The key in our arguments is the following lemma

Lemma 1.1 (see [10]). Assume that is a bounded domain in and the weight satisfies Then the following embedding hold:

(i) continuously for where

(ii) compactly for any

In the sequel we denote by the and the quantities and respectively, where and are induced by condition the . The assumptions concerning the coefficient functions of (1.1) are the following:

(A) and either there exists of positive Lebesgue measure, i.e, such that for all neither in

(D) and either there exists of positive Lebesgue measure, i.e, such that for all neither in

(B) a.e, in and where

Many authors studied the existence of solutions for such problems (equations or systems) ;see for example [5,6,11,15,16,18,19].

Recently in [7], the authors considered the system

They are concerned with the nonexistence and multiplicity of nonnegative, nontrivial solutions.

Also, we mention some results concerning the associated eigenvalue problem. Let be the first eigenvalue of the Diruchlet porblem

[picture] | (1.3)

where the functions and satisfy and the exponents satisfy

[picture] | (1.4)

then, we have that is a positive number, which is characterized variationally by

[picture] | (1.5)

Moreover, is isolated, the associated eigenfunction is componentwise nonnegative and is the only eigenvalue of (1:2) to which corresponds a componentwise nonnegative eigenfunction. In addition, the set of all eigenfunctions corresponding to the principal eigenvalue forms a one-dimentional manifold which is defined by

In the rest of this article, the following assumption is required.

[picture] | (1.6)

The aim of this work is to extend or complete some of the above results for system (1.1). Our assumptions are as follows: are -fonction satisfying the hypotheses below:

(F.1) There exist positive constants such that

with

(F.2) There exist and such that

(F.3) There exists positive constant such that

Next, we introduce the functionals

Lemma 1.2. The functionals I, J, are well defined. Moreover, I is continuous and J are compact.

We say that is a weak solution of problem (1.1) if is a critical point of the functionals

The main results of this paper are the following two theorems.

2. Proof of the Main Result

Theorem 2.1. 2.1. Let (F.1) hold, such that (1.4) satisfait, then there exist such that system (1.1) possesses a weak solution for all

Lemma 2.2. 2.2. Let be a sequence weakly converging to in There we have

(i)

(ii)

Proof. (i) Let be a sequence that converges weakly to By the weak lower semicontinuity of the norm in the space and we deduce that

The compactness of operator J, by lemma 2.2, imply the conclusion.

Lemma 2.3. 2.3. The functional is coercive and bounded from below.

Proof. By (F.1) there exists such that for all and we deduce that

Applying Young’s inequality, we obtain

[picture] | (2.1)

where are the embedding constants of and

Consequently

Tayking such that for all it follows that for is coercive, indeed as .

Proof of theorem 2.1 The coerciveness of and the weak sequential lower semi continuity are enough in order to prove that attaints its infimum, so the system (1.1) has at least one weak solution.

Lemma 2.4. Let be a bounded sequence in such that is bounded and as Then has a convergent subsequence.

Theorem 2.4. 2.5. In addition to (F.2) and (F.3), then there exist such that system (1.1) possesses a weak nontrivial solution for all

Proof. To prove the existence of a weak nontrivial solution we apply a version of the Mountain Pass theorem due to Ambrosetti and Rabinowitz [1]. For this purpose we verify that satisfies:

(i) the mountain pass type geometry,

(ii) the condition.

(i) d’apres

Hence, there exists r > 0, small enough, such that

On the other hand by using (1.4), we have

Thus, we conclude that ther exists large enough, such that for we have and

(ii) According to Lemma (2.4), it is sufficient to prove that the sequence is bounded in

Let be such a sequence, that is, and as We obtain

which shows from (F.2) that

[picture] | (2.2)

Next, we use the following interpolation inequality: let and suppose that for some measurable function we have that

Then

[picture] | (2.3)

We use (2.3) for and we get

[picture] | (2.4)

[picture] | (2.5)

Using (2.2), we obtain

[picture] | (2.6)

and

[picture] | (2.7)

By Lemma 1.1, it follows that

[picture] | (2.8)

and

[picture] | (2.9)

where and On the other hand, by (F.2) and (2.4) -(2.9),we get

Since is bounded and it follows that is bounded in By Lemma 2:4, we obtain that the functional satisfies the condition (compactness condition).

The assumptions of the mountain pass theorem in [3] are satisfied. Then the functional admits a nontrivial critical point in W and thus system (1.1) has a nontrivial weak solution. The proof of Theorem 2.1 is complete.

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