## Boundary Value Problems

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

Boundary Value Problems20122012:137

DOI: 10.1186/1687-2770-2012-137

Accepted: 19 September 2012

Published: 26 November 2012

## Abstract

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.

### Keywords

Brezis-Nirenberg-type problem minimax method

## 1 Introduction and main results

This paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-type problem with singular coefficients
(1)

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
(2)
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:
(3)
for , , that is, the embedding is continuous, where is the weighted space with the norm
On , we can define the energy functional
(4)

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
(5)

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 1.1 Suppose, then
1. (i)

, such that if , problem (1) has a sequence of solutions with and as .

2. (ii)

, such that if , problem (1) has a sequence of solutions with and as .

## 2 Preliminary results

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. (1)
There exists some at most countable set I, a family of distinct points in , and a family of positive numbers such that
(6)

whereis the Dirac-mass of mass 1 concentrated at.
1. (2)
The following inequality holds
(7)

for some family satisfying
(8)
whereto be the best embedding constants, and

In particular, .

Lemma 2.3 Assumeis a (PS) c sequence with, , then
1. (1)

, there exists such that for any , has a convergent subsequence in .

2. (2)

, there exists such that for any , has a convergent subsequence in .

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

For is a (PS) c sequence, then
(9)
(10)
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
1. (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
(11)
taking in (11), we have
(12)
for any . Let in (12), where , then it follows that
(13)
Taking in (13), we have
(14)
Let in (12), then it follows that
(15)
Thus, it implies that
(16)
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 . □

## 3 Existence of infinitely many solutions

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 [68], 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. (1)

and is bounded below.

2. (2)

If , then and .

3. (3)

For any , there exists such that if and , then satisfies (PS) c condition.

4. (4)

for any , there exists such that if and , then satisfies (PS) c condition.

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 3.2 Given, 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 3.3 Let λ, β be 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,

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

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

(1)
School of Science, Jiangnan University
(2)
School of Mathematical Science, Nanjing Normal University
(3)
Institute of Science, PLA University of Science and Technology

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