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
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 ): 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. derived the explicit formula for the extremal functions of the best embedding constant by applying the Bliss lemma . They got a nontrivial solution for problem (5) including the resonant and nonresonant cases by variational methods. He and Zou  studied problem (5) and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In , 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 solutions with and as .
(ii) , such that if , problem (1) has a sequence of solutions with and as .
2 Preliminary results
Suppose that is an open bounded domain with boundary and , . The embedding is compact if , .
Lemma 2.2 (Concentration compactness principle )
Let , , , , and be the space of bounded measures on . Suppose that is a sequence such that
Then there are the following statements:
(1) There exists some at most countable setI, a family of distinct points in , and a family of positive numbers such that
where is the Dirac-mass of mass 1 concentrated at .
(2) The following inequality holds
for some family satisfying
where to be the best embedding constants, and
In particular, .
Lemma 2.3Assume is a (PS)csequence with , , then
(1) , there exists such that for any , has a convergent subsequence in .
(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
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
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 . □
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
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.
(1) and is bounded below.
(2) If , then and .
(3) For any , there exists such that if and , then satisfies (PS)ccondition.
(4) for any , there exists such that if and , then satisfies (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 is such 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 all are critical values of as .
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
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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