Abstract
By energy estimates and by establishing a local (PS) condition, we obtain the multiplicity of solutions to a class of BrezisNirenbergtype problem with singular coefficients via minimax methods and the Krasnoselskii genus theory.
Keywords:
BrezisNirenbergtype problem; minimax method1 Introduction and main results
This paper is concerned with multiple solutions for the semilinear BrezisNirenbergtype problem with singular coefficients
where
The starting point of the variational approach to the problem is the CaffarelliKohnNirenberg
inequality (see [1]): There is a constant
for all
Let
From the boundedness of Ω and the standard approximation arguments, it is easy to
see that (2) holds for any
for
On
From (4), J is well defined in
BreizNirenbergtype 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 pseudoindex theory. In [5], problem (5) has been extended to the pLaplace case by Xuan.
The purpose of this paper is to study the multiplicity of solutions for the BreizNirenbergtype 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
(i)
(ii)
2 Preliminary results
Lemma 2.1[5]
Suppose that
Lemma 2.2 (Concentration compactness principle [5])
Let
Then there are the following statements:
(1) There exists some at most countable setI, a family
where
(2) The following inequality holds
for some family
where
In particular,
Lemma 2.3Assume
(1)
(2)
Proof (1) The boundedness of (PS)_{c} sequence.
For
So, we get
We have the boundedness of
From the concentration compactness principle, there exist nonnegative measures μ, ν and a countable family
(2) Up to a subsequence,
Since
On the other hand,
Note that
taking
for any
Taking
Let
Thus, it implies that
which implies that
Hence,
On the other hand,
then
However, if
Up to now, we have shown that
So, by the BreizLieb lemma,
since
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
Assume that
Define
Then, given
where
Lemma 3.1
(1)
(2) If
(3) For any
(4) for any
Proof (1) and (2) are immediate. To prove (3) and (4), observe that all (PS)_{c} sequences for
Lemma 3.2Given
Proof Fix m and let
Therefore, we can choose
Then
Lemma 3.3Letλ, βbe as in (3) or (4) of Lemma 3.1. Then all
Proof It is clear that
Since
Therefore,
Consequently,
a contradiction, hence
With Lemma 3.1 to Lemma 3.3, we have proved Theorem 1.1.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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).
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