Journal of Applied Mathematics

Volume 2014 (2014), Article ID 482740, 6 pages

http://dx.doi.org/10.1155/2014/482740

Research Article

Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents: Existence and Multiplicity of Nontrivial Solutions

Guanwei Chen

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Received 30 November 2013; Accepted 12 January 2014; Published 19 February 2014

Academic Editor: Tai-Ping Chang

Copyright © 2014 Guanwei Chen. 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.

We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.

1. Introduction and Main Results

Let us consider the following problem:

where denotes the -Laplacian differential operator, is an open bounded domain in    with smooth boundary and , , is the Hardy-Sobolev critical exponent, and is the Sobolev critical exponent. Here, we let

which is equivalent to the usual norm of Sobolev space due to the Poincaré inequality. Let

which is the best Hardy-Sobolev constant.

In the case where and hold, then (1) reduces to the quasilinear elliptic problem:

Gonçalves and Alves [1] have studied (4) in involving , and to obtain existence of positive solutions where , , or and a suitable . We should mention that problem (4) with has been widely studied since Brézis and Nirenberg; see [2–4] and the references therein.

Ghoussoub and Yuan [5] have studied (1) with , where . They obtained a positive solution in the case where , , and (in particular if ) hold. They also obtained a sign-changing solution in the case where , , and (in particular if ) hold. For other relevant papers, see [6–12] and the references therein.

We should mention that the energy functional associated with (1) is defined on , which is not a Hilbert space for . Due to the lack of compactness of the embedding in and , we cannot use the standard variational argument directly. The corresponding energy functional fails to satisfy the classical Palais-Smale ( for short) condition in . However, a local () condition can be established in a suitable range. Then the existence result is obtained via constructing a minimax level within this range and the Mountain Pass Lemma due to Ambrosetti et al. [2] and Rabinowitz [13].

In this paper, we study (1) with a general nonlinearity by using a variational method; besides, we also considerably generalize the results obtained in [5]. In what follows, we always assume that the nonlinearity satisfies . Let , . To state our main results, we still need the following assumptions.

, , and uniformly for .There exists a constant with such that

, , and uniformly for .There exists a constant with such that

Now, our main results read as follows.

Theorem 1. Suppose that , , , and hold. If , , and

hold, then (1) has at least one positive solution.

Theorem 2. Suppose that , , , and hold. If , , and (7) hold, then (1) has at least two distinct nontrivial solutions.

Noting that and imply that (7) holds, therefore, we have the following corollaries.

Corollary 3. Suppose that , , , and . Moreover, and hold; then (1) has at least one positive solution.

Corollary 4. Suppose that , , , and . Moreover, and hold; then (1) has at least two distinct nontrivial solutions.

Remark 5. Theorem 1 generalizes Theorem  1.3 in [5], where the author only studied the special situation that , . There are functions satisfying the assumptions of our Theorem 1 and not satisfying those in [5]. Let

where , , , and . Obviously, satisfies all the conditions of Theorem 1 in this paper, while it does not satisfy the conditions of Theorem  1.3 in [5].

The rest of this paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

2. Preliminaries

In what follows, we let denote the norm in . It is obvious that the values of for are irrelevant in Theorem 1, so we may define

We firstly consider the existence of nontrivial solutions to the problem:

The energy functional corresponding to (10) is given by

By Hardy-Sobolev inequalities (see [5, 14]) and , we know . Now it is well known that there exists a one-to-one correspondence between the weak solutions of (10) and the critical points of on . More precisely we say that is a weak solution of (10), if, for any , there holds

Lemma 6 (see [15]). If a.e. in and for all and some , then

Lemma 7. For any , , and , we have .

Proof. Let

Clearly, , , so .

Lemma 8 (see [5]). If hold, then we have

(i) is independent of , and will henceforth be denoted by ;(ii) is attained when by the functions

for some . Moreover, the functions are the only positive radial solutions of

in , and satisfy

Lemma 9. If , , and hold, then satisfies condition.

Proof. Suppose that is a sequence in . By , we have

where . Hence we conclude that is a bounded sequence in . So there exists ; going if necessary to a subsequence, we have

By the continuity of embedding, we have . From [5], going if necessary to a subsequence, one can get that

as . By , we know that for any there exists such that

Set . When , , we get

It follows from Vitali’s theorem that

Similarly, we can also get

Since , we have

Let , which together with Lemma 6 implies

From (20), we can obtain

Note that as , which together with Lemma 6 implies

Therefore, one gets that

From (26) and (27), we have

then as . Otherwise, there exists a subsequence (still denoted by ) such that

By (3), we have

then . That is, . It follows from (29) and that . However, we have by (27) and (). We get a contradiction. Therefore, we can obtain

From the discussion above, satisfies condition.

In the following, we shall give some estimates for the extremal functions. Let

Define a function such that for , for , , where . Set

so that . Then, by using the argument as used in [5], we can get the following results:

Moreover, by using the Sobolev embedding theorem and (36), one can deduce

Lemma 10. Suppose that . If , and (7) hold, then there exists , , such that

Proof. We consider the functions

Since , , and for small enough, is attained for some . Therefore, we have

and hence

Therefore, we obtain

By (), we can easily get

Hence, we can get

By (36)–(38), when is small enough, we conclude that

On the one hand, from Lemma 7 and (36), it follows that

On the other hand, the function attains its maximum at and is increasing in the interval . Note that () implies , which together with (36), (46), and (47) implies that

Furthermore, from (7) and (37), we get

By (7), we have , which implies

Therefore, by choosing small enough, we have

Hence, the proof of the lemma is completed by taking .

3. Proofs of Main Results

Proof of Theorem 1. Let . From the Sobolev and Hardy-Sobolev inequalities, we can easily get

It follows from () that

uniformly for all . Therefore, we deduce that

for all and for . Then one gets

for all and for . By (52) and (55) we have

for small enough. So there exists such that

By Lemma 10, there exists with such that

It follows from the nonnegativity of that

Therefore, , so we can choose such that

By virtue of the Mountain Pass Lemma in [16], there is a sequence satisfying

where

Note that

By Lemma 9 we can assume that in . From the continuity of , we know that is a weak solution of problem (10). Then , where . Thus . Therefore, is a nonnegative solution of (1). By the Strong Maximum Principle [17], is a positive solution of problem (1). Therefore, Theorem 1 holds.

Proof of Theorem 2. By Theorem 1, we know that (1) has a positive solution . Set

It follows from Theorem 1 that the equation

has at least one positive solution . Let ; then is a solution of

It is obvious that , , and . So (1) has at least two nontrivial solutions. Therefore, Theorem 2 holds.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the referees and the editors for their helpful comments and suggestions. Research is supported by the Tianyuan Fund for Mathematics of NSFC (Grant no. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant no. 13A110015).

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