Existence Results for a Class of Quasilinear Elliptic Systems Different Weights
1Department of Mathematics, University Tebessa, Tebessa, Algeria
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
International Journal of Partial Differential Equations and Applications, 2014 2 (2),
Received March 29, 2014; Revised April 26, 2014; Accepted April 27, 2014Copyright: © 2014 Science and Education Publishing. All Rights Reserved.
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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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  and the references therein. The key in our arguments is the following lemma.
Lemma 1.1 (see ). 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
Recently in , 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
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
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
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  For this purpose we verify that satisfies:
(i) the mountain pass type geometry,
(ii) the condition.
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
We use for and we get
Using , we obtain
By Lemma 1.1, it follows that
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  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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