Abstract and Applied AnalysisVolume 2012 (2012), Article ID 579481, 19 pagesdoi:10.1155/2012/579481

Research Article

Multiplicity of Positive Solutions for Weighted Quasilinear Elliptic Equations Involving Critical Hardy-Sobolev Exponents and Concave-Convex Nonlinearities

Tsing-San Hsu and Huei-Li Lin

Center for General Education, Chang Gung University, Kwei-Shan, Taiwan

Received 24 September 2011; Revised 20 December 2011; Accepted 16 January 2012

Academic Editor: Martin D. Schechter

Copyright © 2012 Tsing-San Hsu and Huei-Li Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By variational methods and some analysis techniques, the multiplicity of positive solutions is obtained for a class of weighted quasilinear elliptic equations with critical Hardy-Sobolev exponents and concave-convex nonlinearities.

1. Introduction and Main Results

Let be a smooth bounded domain in and . We will study the multiplicity of positive solutions for the following quasilinear elliptic problem: where , , , , , , , is the critical Sobolev-Hardy exponent. Note that is the critical Sobolev exponent.

In this paper, denotes the space obtained as the completion of with respect to the norm . The energy functional of (1.1) is defined on by Then . is said to be a solution of (1.1) if for all and a solution of (1.1) is a critical point of . By the standard elliptic regularity argument, we deduce that .

Problem (1.1) is related to the following Hardy inequality [1]: which is also called the (general or weighted) Hardy-Sobolev inequality. For the sharp constants and extremal functions, see [2, 3]. If , then and the following (general or weighted) Hardy inequality holds [1, 4]: where is the best Hardy constant.

In the space , we employ the following norm if : By (1.4) it is equivalent to the usual norm of the space . According to (1.4), we can define the following best constant for , and : From Kang [5, Lemma  2.2], is independent of . Thus, we will simply denote that .

When , we set and , then (1.1) is equivalent to the following quasilinear elliptic equations: where , , , , and .

Such kind of problem relative with (1.7) has been extensively studied by many authors. When , people have paid much attention to the existence of solutions for singular elliptic problems (see [6–16] and their references therein), besides, in the most of these papers, the operator with Sobolev-Hardy critical exponents (the case that ) has been considered. Some authors also studied the singular problems with Sobolev-Hardy critical exponents (the case that ) (see [17–22] and their references therein). In [23, 24], the authors deal with doubly-critical exponents.

When . The quasilinear problems related to Hardy inequality and Sobolev-Hardy inequality have been studied by some authors [25–32]. Here we recall the work in [25], where the extremal functions for the best Sobolev constant were studied. The results can be employed to study the problems with critical Sobolev exponents and Hard terms, see [25, 28]. In [26] it is investigated in a quasilinear elliptic equation involving doubly critical exponents by the concentration compactness principle [33, 34].

We should note that the nonlinearities of problems studied in [11–14, 26, 28, 31] are all not sublinear or -sublinear near the origin. To the best of our knowledge, there are few results of problem (1.7) with nonlinearities being -sublinear near the origin for . We are only aware of the works [20, 30, 32] which studied the existence and multiplicity of positive solution of problem (1.7) with . In this paper, we study (1.1) and extend the results of [20, 30, 32] to the case and .

For , , and , consider the following limiting problem: where is the space obtained as the completion of with respect to the norm . From [5, Lemma  2.2], we know (1.8) has a unique ground state solution satisfying and all ground states must be of the form for some , that is, is achieved by . Moreover, is radially symmetric and possesses the following properties: where are positive constants and , are the zeros of the function which satisfy . Furthermore, there exist the positive constants and such that

Throughout this paper, let be the positive constant such that , where . By Hölder and Sobolev-Hardy inequalities, for all , we obtain Set where is the volume of the unit ball in .

Furthermore, from and , we can deduce that which implies Combining these with , we get .

We are now ready to state our main results.

Theorem 1.1. Assume that , , , , and . Then one has the following results. (i)If , then (1.1) has at least one positive solution in .(ii)If , then (1.1) has at least two positive solutions in .

Remark 1.2. In [5], Kang considered (1.1) with -sublinear perturbation of . Via variational methods, he proved the existence of positive solutions of (1.1) when the parameters satisfy suitable conditions. But the existence of positive solutions for (1.1) involving the -sublinear of is not considered. In this paper, we will give a complement result.This paper is organized as follows. In Sections 2 and 3, we give some preliminaries and some properties of Nehari manifold. In Section  4, we complete proofs of Theorem 1.1. At the end of this section, we explain some notations employed. In the following discussions, we will denote various positive constants as , and omit in the integral for convenience. We denote as a ball centered at the origin with radius , and is the volume of the unit ball in . We denote the norm in by for , and is the closure of with the norm . denoting the dual space of . denotes , and means as . By we always mean it is a generic infinitesimal value.

2. Nehari Manifold

Since the functional is not bounded from below on , we will work on Nehari manifold. For we define We recall that any nonzero solutions of (1.1) belong to . Moreover, by definition, we have that if and only if The following result is concerned with the behavior of on .

Lemma 2.1. is coercive and bounded from below on .

Proof. If , then by (1.14) and (2.2), we get Since , and , it follows that is coercive and bounded from below on .

Define , by , that is, Note that is of class with Furthermore, if , then by (2.2), we have that Now we split into three sets:

The following result shows that minimizers on are the “usual” critical points for .

Lemma 2.2. Suppose is a local minimizer of on and . Then, in .

Proof. See [30, Lemma  2.2].

Motivated by the above result, we will get conditions for .

Lemma 2.3. for all .

Proof. We argue by contradiction. Suppose that there exists a such that . Let be arbitrary, then by (2.2), (2.7), and (2.8), we have that By (1.14), (2.10), and Sobolev-Hardy inequality, we get Hence we must have which is a contradiction.

For each , let

Lemma 2.4. If , then for each , the set intersects exactly twice. More specifically, there exist a unique such that and a unique such that . Moreover, and

Proof. The proof is similar to that of [29, Lemma  2.7] and is omitted.

We remark that by Lemma 2.3 we have, for all . Furthermore, by Lemma 2.4 it follows that and are non-empty and by Lemma 2.1 we may define

Theorem 2.5. (i)If , then one has .(ii)If , then for some positive constant . In particular, for each , one has .

Proof. (i) Let . By (2.7), and so also using (2.2), Therefore, from the definition of and , we can deduce that .(ii) Let . By (2.7), Moreover, by Sobolev-Hardy inequality, This implies By (2.4) and (2.20), we have Thus, if , then for some positive constant .

Remark 2.6. If , then by Lemma 2.4 and Theorem 2.5, for each , we can easily deduce that

3. Proof of the Main Results

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values, and (PS)-conditions in for as follows.

Definition 3.1. (i) For , a sequence is a -sequence in for if and strongly in as .(ii) is a (PS)-value in for if there exists a -sequence in for .(iii) satisfies the -condition in if any -sequence in for contains a convergent subsequence.Now, we use the Ekeland variational principle [35] to get the following results.

Proposition 3.2. (i) If , then has a -sequence . (ii) If , then has a -sequence .

Proof. The proof is similar to [29, Proposition  3.3] and the details are omitted.

Now, we establish the existence of a local minimum for on .

Theorem 3.3. Assume that , , , , and . If , then there exists such that (i),(ii) is a positive solution of (1.1),(iii) as .

Proof. By Proposition 3.2(i), there exists a minimizing sequence such that Since is coercive on (see Lemma 2.1), we get that is bounded in . From [5, Lemma  2.1], we deduce that the embedding is compact for . Thus, there exists , passing to a subsequence if necessary, using similar arguments found in [27, 36], then one can get that as Consequently, passing to the limit in , by (3.1) and (3.2), as , we have for all . That is, . Thus is a weak solution of (1.1). Furthermore, from and (2.3), we deduce that Let in (3.4), by (3.1) and (3.2), and since by (i) of Theorem 2.5, we get Thus , and since , it follows that and, in particular, .Next, we will show, up to a subsequence, that strongly in and . From the fact , (2.3) and the Fatou’s lemma, it follows that which implies that and . Standard argument shows that strongly in . Moreover, . Otherwise, if , by Lemma 2.4, there exist unique and such that , and . Since there exists such that . By Lemma 2.4 we get that which is a contradiction. Since and , by Lemma 2.2, we may assume that is a nontrivial nonnegative solution of (1.1). By [5, Lemma  2.3], it follows that in . Finally, by (1.14) and (2.8), we obtain which implies that as .

Next, we will establish the existence of the second positive solution of (1.1) by proving that satisfies the -condition.

Lemma 3.4. Let be a bounded sequence in . If is a -sequence for with then there exists a subsequence of converging weakly to a nonzero solution of (1.1).

Proof. Let be a -sequence for with . Since is bounded in and the embedding is compact for (see [5, Lemma  2.1]), thus passing to a subsequence if necessary, we may assume that as Using the same argument in Theorem 3.3, we deduce that and Next we verify that . Arguing by contradiction, we assume . Set Since and is bounded in , then by (3.12), we can deduce that that is, If , then by (3.12)–(3.15), we get which contradicts . Thus we conclude that . Furthermore, the Sobolev-Hardy inequality implies that Then as , we have , which implies that Hence, from (3.12)–(3.18) we get This contradicts the definition of . Therefore, is a nontrivial solution of (1.1).

Lemma 3.5. If , , , , and , then for any , there exists such that In particular, for all .

Proof. Let be a ground state solution of (1.8), small enough such that , , , for for . Set and . Then, following the same lines as in [5], we get the following estimates as : where is given in the introduction satisfying .Now we consider the following functions: Using the definitions of and , we get Combining this with (3.21), let , then there exists independent of such that On the other hand, by the fact for and by (3.21) and (3.22), we can get that Hence as , , by (3.28) we have that (i) If , then by (3.23) we have that and since , then Combining this with (3.26) and (3.29), for any , we can choose small enough such that (ii) If , then by (3.23) and we have that Combining this with (3.26) and (3.29), for any , we can choose small enough such that From (i) and (ii), (3.20) holds by taking .From Lemma 2.4, the definition of , and (3.20), for any , we obtain that there exists such that and The proof is thus complete.

Now, we establish the existence of a local minimum of on .

Theorem 3.6. Assume that , , , and . If , then there exists such that (i),(ii) is a positive solution of (1.1).

Proof. If , then by Theorem 2.5 (ii), Proposition 3.2 (ii), and Lemma 3.5, there exists a -sequence in for with . Since is coercive on (see Lemma 2.1), we get that is bounded in . From Lemma 3.4, there exists a subsequence still denoted by and a nontrivial solution of (1.1) such that weakly in .First, we prove that . Arguing by contradiction, we assume . Since is closed in , we have . Thus, by Lemma 2.4, there exists a unique such that . If , then it is easy to see that From Remark 2.6, , , and (3.36), we can deduce that This is a contradiction. Thus, .Next, by the same argument as that in Theorem 3.3, we get that strongly in and for all . Since and , by Lemma 2.2, we may assume that is a nontrivial nonnegative solution of (1.1). Finally, by [5, Lemma  2.3], we obtain that is a positive solution of (1.1).

Now, we complete the proof of Theorem 1.1. The part (i) of Theorem 1.1 immediately follows from Theorem 3.3. When , by Theorems 3.3 and 3.6, we obtain (1.1) has two positive solutions and such that , . Since , this implies that and are distinct. This completes the proof of Theorem 1.1.

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