Existence Results for a Class of Quasilinear Elliptic Systems Different Weights

Tahar Bouali, Rafik Guefaifia

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Existence Results for a Class of Quasilinear Elliptic Systems Different Weights

Tahar Bouali1,, Rafik Guefaifia1

1Department of Mathematics, University Tebessa, 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.

Cite this article:

  • Bouali, Tahar, and Rafik Guefaifia. "Existence Results for a Class of Quasilinear Elliptic Systems Different Weights." International Journal of Partial Differential Equations and Applications 2.2 (2014): 27-31.
  • Bouali, T. , & Guefaifia, R. (2014). Existence Results for a Class of Quasilinear Elliptic Systems Different Weights. International Journal of Partial Differential Equations and Applications, 2(2), 27-31.
  • Bouali, Tahar, and Rafik Guefaifia. "Existence Results for a Class of Quasilinear Elliptic Systems Different Weights." International Journal of Partial Differential Equations and Applications 2, no. 2 (2014): 27-31.

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1. Introduction

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

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

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:

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

and either there exists of positive Lévesque measure, i.e., such that , for all , neither in

a.e, in and where

Many authors studied the existence of solutions for such problems 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 problem

where the functions and satisfy , and the exponents satisfy

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

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

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

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 function satisfying the hypotheses below:

There exist positive constants such that

with

There exist and such that

There exists positive constant such that

Next, we introduce the functionals

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

We say that is a weak solution of problem (1,1) if is a critical point of the functionals

i.e

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

2. Proof of the Main Result

Theorem 2.1. Let hold, such that satisfait, then there exist such that system possesses a weak solution for all

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

Proof Let be a sequence that converges weakly to By the weak lower semi continuity of the norm in the space and we deduce that

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

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

Proof By there exists such that for all and a.e. we deduce that

Applying Young's inequality, we obtain

where are the embedding constants of and

Consequently

Taking 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 2.5. In addition to and , 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.

(iii) d'apres

Hence, there exists small enough, such that

On the other hand by using we have

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

(ii) According to Lemma , 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 that

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

Then

We use for and we get

Using , we obtain

and

By Lemma 1.1, it follows that

and

where and On the other hand, by and - , 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 is complete.

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