Existence of Solutions for Degenerate Elliptic Problems in Weighted Sobolev Space

Abstract

This paper is devoted to the study of the existence of solutions to a general elliptic problem , with and , where is a Leray-Lions operator from a weighted Sobolev space into its dual and is a nonlinear term satisfying , , and a growth condition with respect to . Here, , are weight functions that will be defined in the Preliminaries.

1. Introduction

Let be a bounded domain in and let be a real number with . Denote by the weighted Sobolev space , associated with a vector of weight functions , which is endowed with the usual norm . In this paper, we consider a general class of degenerate elliptic problems:where and the right-hand side term , where , . We also assume the following:(H1)The expressionis a norm defined on and it is equivalent to .(H2)There exist a weight function and a parameter , , such thatwith . The Hardy inequalityholds for every with a constant independent of . Moreover, the embeddingis compact. Interested reader may refer to [1] for some examples of weights which satisfy the above Hardy inequality (see (4)).(H3) is a Carathéodory vector-valued function, and for all , there holdwhere is a positive function in , , and the constants , are both positive.(H4)Let be a Carathéodory function satisfying the following assumptions:for some , andwith , a continuous increasing function, and , a nonnegative function in .

In the past decade, much attention has been devoted to nonlinear elliptic equations because of their wide application to physical models such as non-Newtonian fluids, boundary layer phenomena for viscous fluids, and chemical heterogenous model. When , Akdim et al. [2] proved in the variational setting, under assumptions (H1)–(H4), that, for every , with satisfying the sign conditionProblem (1) has a solution , where the authors used the approach based on the strong convergence of the positive part (negative part ) of (the approximating sequence of ). Ammar [3] extended this existence result to problems with general data , under hypotheses . They also used a similar approach to prove the existence of renormalized solutions. When , Aharouch et al. [4] proved the existence result for problem (1), by assuming the sign condition (11). For more details on weighted Sobolev spaces, the readers may refer to [5].

Boccardo et al. [6] considered the nonlinear boundary value problem where and with sign condition (9) for large values of . By combining the truncation technique with some delicate test functions, the authors showed that the problem has a solution . Mainly motivated by [4, 6], we investigate the elliptic problem (1) in weighted Sobolev space. By choosing test functions different from those employed in [4, 6], we show that problem (1) admits at least one weak solution with (9) instead of the sign condition (11). It is worth pointing out that (9) gives a sign condition on only for large values of , which brings about many difficulties. The essential one of those is that we have to construct some new test functions to obtain the a priori estimates of the approximation solutions since the usual one is not a proper test function for our problem. The outline of this paper is as follows. In Section 2, we give some preliminaries and some technical lemmas. The main results will be stated and proved in Section 3.

2. Preliminaries

In this section, we give some preliminaries (see [5]). Throughout this section, we assume that the vector field satisfies assumptions (6)–(8) and satisfies (9)-(10). Let be a vector of measurable weight functions strictly positive a.e. in , such thatWe define the weighted space with weight on asWith this space, we equip the normWe denote by the space of all real-valued functions such that the derivatives (see [5]) in the sense of distributions satisfyendowed with the normLet be the closure of with respect to the norm . Then, is a reflexive Banach space whose dual is equivalent to , where ,  , and . As usual, for , in , with , we denote and .

The following lemmas will be needed throughout this paper (refer to [2, 7]).

Lemma 1. Let and be two nonnegative real numbers, and letwith . Then,

Lemma 2. Let and , with , . If in , then weakly in , where is a weight function on .

Lemma 3 (assume ). Let be uniformly Lipschitzian, with . Let . Moreover, if the set of discontinuity points of is finite, then

Lemma 4 (assume ). Let and , , be the usual truncation. Then, . Moreover, one has

Lemma 5 (assume and ). Let be a sequence of functions in such that weakly in andThen, strongly in .

3. Main Results

Firstly, we give the definition of weak solution for problem (1).

Definition 6. One says is a weak solution to problem (1), provided that

Now, we will state and prove our main result on the existence of weak solutions to problem (1).

Theorem 7. Let be in and . Then, there exists at least one solution to problem (1).

Proof. The proof will be divided into 5 steps.
Step 1 (the approximation equation). We introduce the following approximation equation of problem (1). Let be a sequence of functions that converges to strongly in and let ,then is bounded and satisfies (10) andfor almost every in , for every in , and for every in with . By the results of [2], there exists a solution ofwhich satisfiesfor every .
Step 2 (the weak convergence in ). Take as a test function in (27), where is defined in (9) and is as in (19). Writing and for simplicity, we haveThanks to Young’s inequality and (7), we haveSince is bounded in and , it follows from the above inequality thatwhere is independent of . Splitting the second term on the left-hand side where and , we can writeUsing (9) and (10), we getHence,Recalling (19) in Lemma 1, let , ; we then obtainwhich impliesor equivalentlywhere is some positive constant. Therefore, we can extract a subsequence, still denoted by itself, such thatBy (5) and (37), we have for a subsequence strongly in and a.e. in . Then, is bounded in . Hence, by the results of [8], we haveStep 3 (the strong convergence in ). For every , we will prove that converges strongly to in . We first prove thatHere, we denote by the set of natural numbers. Choosing as a test function in (27) with , using (7) and Young’s inequality, we obtainNoticing (9) and that has the same sign as if and is zero if , we getDropping the nonnegative term, we haveSincewe obtainTaking into account the fact that is compact in and , we deduce thatHence,Noticing that and (9), this completes the proof of assertion (39).
Let be fixed, , and choose as a test function in (27), where is defined in Lemma 1 (refer to [8–10]). We thus obtainIn the following, represents a quantity which converges to zero as firstly and secondly . For convenience, we writeObserve that, in the weak topology of and almost everywhere in , we haveNow, as is compact in and (49), we haveThanks to weakly in , , and (49), we obtainwhere . We can decompose asOwing to , we getSince is zero whenever is not zero, hence,Since on the set , we see that, as ,As , Lebesgue’s dominated convergence theorem guarantees thatBy (6), (49), and the fact that is bounded in , we obtainNow we split intoWe will prove thatIn fact,where , are positive constants. Since weakly in and is compact, then strongly in and a.e. in . Hence,Then, by the generalized Lebesgue dominated convergence theorem, we deduce (59). By weakly in and (49), we haveUsing (57) and (62), we haveAs for the term , we decompose it asIt is clear that on the set we getwhile on the set we getBy (9) and the fact that , we obtainUsing (7) and (10) and noticing that , we haveBy the weak convergence of in , (49), and (59), we haveSince , is bounded in and (49),We haveInvoking (63), we haveConsequently, by (19) and letting and , it yieldsSince , we haveTogether with Lemma 5 and the assumptions on , we obtainwhich in turn impliesFor any measurable set of , we haveLet . Thanks toby (39), there exists such thatWhile is fixed, we getOwing to the strong compactness of in , there exists such that if , thenHence,Thus, the sequence is equi-integrable. Thanks to Vitali theorem, the equi-integrability together with (76) implies that converges strongly to in .
Step 4 (the strong convergence in ). Note that (76) implies thatOn the other hand, for any measurable set of , we haveLet be fixed. We haveChoose in (46) such thatBy (10), we haveSince belongs to and is compact in , there exists such that if , thenThus, we have proved that is equi-integrable. Invoking (86) and (88) and by Vitali theorem,Step 5 (passing to the limit). Now, by passing to the limit in (27), we obtainthat is, is a weak solution to problem (1). The proof is complete.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally to the paper and read and approved its final version.

Acknowledgment

This study was supported by the National Science Foundation of China (11271154, 11401252).