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
The starting point of the variational approach to the problem is the Caffarelli-Kohn-Nirenberg inequality (see ): There is a constant such that
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.
2 Preliminary results
Lemma 2.2 (Concentration compactness principle )
Then there are the following statements:
(2) The following inequality holds
Proof (1) The boundedness of (PS)c sequence.
So, we get
Thus, it implies that
which implies that
On the other hand,
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,
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
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
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
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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