Abstract and Applied Analysis

Volume 2012 (2012), Article ID 806397, 15 pages

doi:10.1155/2012/806397

Research Article

Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

Zonghu Xiu

Science and Information College, Qingdao Agricultural University, Qingdao 266109, China

Received 11 October 2012; Revised 1 November 2012; Accepted 5 November 2012

Academic Editor: Jifeng Chu

Copyright © 2012 Zonghu Xiu. 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 consider the existence of multiple solutions of the singular elliptic problem , as , where , , , , , , . By the variational method and the theory of genus, we prove that the above-mentioned problem has infinitely many solutions when some conditions are satisfied.

1. Introduction and Main Results

In this paper, we consider the existence of multiple solutions for the singular elliptic problem

where . and are nonnegative functions in .

In recent years, the existence of multiple solutions on elliptic equations has been considered by many authors. In [1], Assunção et al. considered the following quasilinear degenerate elliptic equation:

where . When , where and ; the authors proved that problem (1.2) has at least two positive solutions. Rodrigues in [2] studied the following critical problem on bounded domain :

By the variational method on Nehari manifolds [3, 4], the author proved the existence of at least two positive solutions and the nonexistence of solutions when some certain conditions are satisfied. When and , Miotto and Miyagaki in [5] considered the semilinear Dirichlet problem in infinite strip domains

The authors also proved that problem (1.4) has at least two positive solutions by the methods of Nehari manifold. For other references, we refer to [6–11] and the reference therein. In fact, motivated by [1, 2, 5], we consider the problem (1.1). Since our problem is singular and is studied in the whole space , the loss of compactness of the Sobolev embedding renders a variational technique that is more delicate. By the variational method and the theory of genus, we prove that problem (1.1) has infinitely many solutions when some suitable conditions are satisfied.

In order to state our result, we introduce some weighted Sobolev spaces. For and in , we define the spaces and as being the set of Lebesgue measurable functions , which satisfy

Particularly, when , we have

We denote the completion of by with the norm of

where and . It is easy to find that is a reflexive and separable Banach space with the norm .

The following Hardy-Sobolev inequality is due to Caffarelli et al. [12], which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants such that

where is called the Sobolev critical exponent.

In the present paper, we make the following assumptions:

for , where ; for , where . for , where .

Then, we give some basic definitions.

Definition 1.1. is said to be a weak solution of (1.1) if for any there holds

Let be the energy functional corresponding to problem (1.1), which is defined as

for all . Then the functional and for all , there holds

It is well known that the weak solutions of problem (1.1) are the critical points of the functional , see [13]. Thus, to prove the existence of weak solutions of (1.1), it is sufficient to show that admits a sequence of critical points in .

Our main result in this paper is the following.

Theorem 1.2. Let , , , . Assume – are fulfilled. Then problem (1.1) has infinitely many solutions in .

2. Preliminary Results

Our proof is based on variational method. One important aspect of applying this method is to show that the functional satisfies condition which is introduced in the following definition.

Definition 2.1. Let and be a Banach space. The functional satisfies the condition if for any such that

contains a convergent subsequence in .

The following embedding theorem is an extension of the classical Rellich-Kondrachov compactness theorem, see [14].

Lemma 2.2. Suppose is an open bounded domain with boundary and . . Then the embedding is continuous if and , and is compact if and .

Now we prove an embedding theorem, which is important in our paper.

Lemma 2.3. Assume - and . Then the embedding is compact.

Proof. We split our proof into two cases.

(i) Consider .

By the Hölder inequality and (1.9) we have that

where . Then the embedding is continuous. Next, we will prove that the embedding is compact.

Let be a ball center at origin with the radius . For the convenience, we denote by , that is, . Assume is a bounded sequence in . Then is bounded in . We choose in Lemma 2.2, then there exist and a subsequence, still denoted by , such that as . We want to prove that

where . In fact, we obtain from (2.2) that

The fact shows that

Then (2.4) and (2.5) imply that

which gives (2.3).

In the following, we will prove that strongly in .

Since is a reflexive Banach space and is bounded in . Then we may assume, up to a subsequence, that

In view of (2.3), we get that for any there exists large enough such that

On the other hand, due to the compact embedding in Lemma 2.2, we have that

Therefore, there is such that

for . Thus, the inequalities (2.8) and (2.10) show that

This shows that is convergent in .

(ii) Consider .

It follows from (1.8) and the Hölder inequality that

where . Thus, the fact of and (2.12) imply that the embedding is continuous. Similar to the proof of (i) we can also prove that the embedding is compact for .

Similarly, we have the following result of compact embedding.

Lemma 2.4. Assume and , then the embedding is compact.

The following concentration compactness principle is a weighted version of the Concentration Compactness Principle II due to Lions [15–18], see also [19, 20].

Lemma 2.5. Let . Suppose that is a sequence such that

where are measures supported on and is the space of bounded measures in . Then there are the following results.

(1) There exists some at most countable set , a family of distinct points in , and a family of positive numbers such that

where is the Dirac measure at .(2) The following equality holds

for some family satisfying

(3)There hold

where

Lemma 2.6. Let . Then satisfies the condition with , where is as in (1.8).

Proof. We will split the proof into three steps.

Step  1. is bounded in .

Let be a sequence of in , that is,

Then, we have

Since , (2.20) shows that is bounded in .

Step  2. There exists in such that in .

The inequality (1.8) shows that is bounded in . Then the above argument and the compactness embedding in Lemma 2.2 mean that the following convergence hold:

It follows from Lemma 2.5 that there exist nonnegative measures and such that

Thus, in order to prove it is sufficient to prove that .

For the proof of , we define the functional such that

where belongs to the support of . It follows from (2.1) that

Since is bounded, we can get from (1.8)-(1.9), Lemmas 2.3 and 2.5 that

On the other hand,

where . Then ; furthermore, (2.16) implies that or . We will prove that the later does not hold. Suppose otherwise, there exists some such that . Then (2.19) and Lemma 2.4 show that

which contradicts the hypothesis of . Then .

Similarly, we define the functional as

Then, the similar proof as above shows that . Thus, we can deduce from (2.22) that

which implies that in .

Step  3. converges strongly in .

The following inequalities [21] play an important role in our proof:

Our aim is to prove that is a Cauchy sequence of . In fact, let in (1.12), it follows from (2.19) that

where

Using the inequalities (2.31), we can get by direct computation that

with some constant , independent of and .

Then the Hölder inequality together with (1.8) and (2.30) yield that

Similarly, we have from the Hölder inequality, Lemmas 2.3 and 2.4 that

Therefore, the above estimates imply that , that is, is a Cauchy sequence of . Then converges strongly in and we complete the proof.

Similarly, we have the following lemma.

Lemma 2.7. Let . Then satisfies the condition with , where are as in (1.8), and (1.9) respectively.

Proof. Step  1. is bounded in .

Let be a sequence of in . Then we have from Lemma 2.3 that

Since , (2.37) shows that is bounded in .

Step  2. There exists in such that in .

Similar to the proof of Lemma 2.5, we can get that or by applying the functional . Now we prove that there is no such that . Suppose otherwise, then

Let

Then has the unique minimum point at

Then it follows from (2.38) that

which contradicts the hypothesis of .

Step  3. converges strongly in .

By Lemma 2.4, this result can be similarly obtained by the method in Lemma 2.6, so we omit the proof.

3. Existence of Infinitely Solutions

In this section, we will use the minimax procedure to prove the existence of infinity many solutions of problem (1.1). Let denotes the class of such that is closed in and symmetric with respect to the origin. For , we recall the genus which is defined by

If there is no mapping as above for any , then , and . The following proposition gives some main properties of the genus, see [13, 22].

Proposition 3.1. Let . Then

(1)if there exists an odd map , then ,(2)if , then ,(3).(4)if is a sphere centered at the origin in , then ,(5)if is compact, then and there exists such that and , where .

Lemma 3.2. Assume (A₁)–(A₃). Then for any , there exists such that

Proof. For given , let be a -dimensional subspace of . If , then for we have

The fact that all the norms on finite dimensional space are equivalent implies that for all

for some constant . Then there exist large and small such that

Denote

Then is a sphere centered at the origin with radius of and

Therefore, Proposition 3.1 shows that .

If , we have

Since is also a norm and all norms on the finite dimensional space are equivalent, we have

Then there exist large and small such that

Denote

Then is a sphere centered at the origin with radius of and

Therefore, Proposition 3.1 shows that .

Let . It is easy to check that . We define

It is not difficult to find that

and for any since is coercive and bounded below. Furthermore, we define the set

Then, is compact and we have the following important lemma, see [22].

Lemma 3.3. All the are critical values of . Moreover, if , then .

Proof of Theorem 1.2. In view of Lemmas 2.6 and 2.7, satisfies the condition in . Furthermore, as the standard argument of [13, 22, 23], Lemma 3.3 gives that has infinity many critical points with negative values. Thus, problem (1.1) has infinitely many solutions in , and we complete the proof.

Acknowledgments

The author would like to express his sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.

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