Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

By energy estimates and by establishing a local (PS) condition, we obtain the multiplicity of solutions to a class of Brezis-Nirenberg-type problem with singular coefficients via minimax methods and the Krasnoselskii genus theory.

This paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-type problem with singular coefficients

where is a bounded smooth domain, and , , , , , , . , are two real parameters.

The starting point of the variational approach to the problem is the Caffarelli-Kohn-Nirenberg inequality (see [1]): There is a constant such that

for all , where

Let be the completion of with respect to the weighted norm defined by

From the boundedness of Ω and the standard approximation arguments, it is easy to see that (2) holds for any in the sense:

for , , that is, the embedding is continuous, where is the weighted space with the norm

On , we can define the energy functional

From (4), J is well defined in , and . Furthermore, the critical points of J are weak solutions of problem (1).

Breiz-Nirenberg-type problems have been generalized to many situations such as

Xuan et al.[2] derived the explicit formula for the extremal functions of the best embedding constant by applying the Bliss lemma [3]. They got a nontrivial solution for problem (5) including the resonant and nonresonant cases by variational methods. He and Zou [4] studied problem (5) and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In [5], problem (5) has been extended to the p-Laplace case by Xuan.

The purpose of this paper is to study the multiplicity of solutions for the Breiz-Nirenberg-type problem (1) with the aid of a minimax method. We obtain multiple nontrivial solutions of (1) by proving the local (PS) condition and energy estimates.

Our main results are the following.

Theorem 1.1Suppose, then

(i) , such that if, problem (1) has a sequence of solutionswithandas.

(ii) , such that if, problem (1) has a sequence of solutionswithandas.

Lemma 2.1[5]

Suppose thatis an open bounded domain withboundary and, . The embeddingis compact if, .

Lemma 2.2 (Concentration compactness principle [5])

Let, , , , andbe the space of bounded measures on. Suppose thatis a sequence such that

Then there are the following statements:

(1) There exists some at most countable setI, a familyof distinct points in, and a familyof positive numbers such that

whereis the Dirac-mass of mass 1 concentrated at.

(2) The following inequality holds

for some familysatisfying

whereto be the best embedding constants, and

In particular, .

Lemma 2.3Assumeis a (PS)_csequence with, , then

(1) , there existssuch that for any, has a convergent subsequence in.

(2) , there existssuch that for any, has a convergent subsequence in.

Proof (1) The boundedness of (PS)_c sequence.

For is a (PS)_c sequence, then

So, we get

We have the boundedness of for , then there exists a subsequence, we still denote it by , such that

From the concentration compactness principle, there exist nonnegative measures μ, ν and a countable family such that

(2) Up to a subsequence, in .

Since is bounded in , we may suppose, without loss of generality, that there exists such that

On the other hand, is also bounded in and

Note that

taking in (11), we have

for any . Let in (12), where , then it follows that

Taking in (13), we have

Let in (12), then it follows that

Thus, it implies that

which implies that

Hence, if .

On the other hand,

then , so that

However, if is given, we can choose so small that for every , the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if is given, we can take so small that for every , the last term on the right-hand side above is greater than 0. Then for each i.

Up to now, we have shown that

So, by the Breiz-Lieb lemma,

since . Thus, we prove that strongly converges to u in . □

In this section, we use the minimax procedure to prove the existence of infinitely many solutions. Let Σ be the class of subsets of , which are closed and symmetric with respect to the origin. For , we define the genus by

Assume that , then we obtain

Define

Then, given , there exists so small that for every , there exists such that for , for , for . Similarly, given , we can choose with the property that , as above exist for each . Clearly, . Following the same idea as in [6-8], we consider the truncated functional

where , and is a nonincreasing function such that if and if . The main properties of are the following.

Lemma 3.1

(1) andis bounded below.

(2) If, thenand.

(3) For any, there existssuch that ifand, thensatisfies (PS)_ccondition.

(4) for any, there existssuch that ifand, thensatisfies (PS)_ccondition.

Proof (1) and (2) are immediate. To prove (3) and (4), observe that all (PS)_c sequences for with must be bounded. Similar to the proof of Lemma 2.3, there exists a convergent subsequence. □

Lemma 3.2Given, there issuch that

Proof Fix m and let be an m-dimensional subspace of . Take , , write with , and . Thus, for , since all the norms are equivalent, we have

Therefore, we can choose so small that . Let , then . Hence, . Denote and let

Then because and is bounded from below. □

Lemma 3.3Letλ, βbe as in (3) or (4) of Lemma 3.1. Then allare critical values ofas.

Proof It is clear that , . Hence, . Moreover, since all are critical values of , we claim that . If , because is compact and , it follows that and there exists such that . By the deformation lemma there exist () and an odd homeomorphism η such that

Since is increasing and converges to , there exists such that and and there exists such that . By the properties of γ, we have

Therefore,

Consequently,

a contradiction, hence . □

With Lemma 3.1 to Lemma 3.3, we have proved Theorem 1.1.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

Project is supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), the China Scholarship Council, Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK2012109), the Fundamental Research Funds for the Central Universities (No. JUSRP11118, JUSRP211A22) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

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