Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential
© Hsu; licensee Springer 2012
Received: 12 March 2012
Accepted: 3 October 2012
Published: 17 October 2012
In this paper, we consider singular elliptic systems involving a strongly coupled critical potential and concave nonlinearities. By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.
MSC: 35J60, 35B33.
KeywordsPalais-Smale condition Nehari manifold strongly coupled elliptic system critical potential
1 Introduction and main results
where is a smooth bounded domain such that , , is the critical Sobolev exponent, is the best Hardy constant and denotes the completion of with respect to the norm and is defined as the completion of the with respect to the norm defined by for .
Definitions of strongly and weakly coupled terms are as follows.
The terms and () are weakly coupled, () is strongly coupled when L or K is a derivative operator. Thus, is strongly coupled when and are positive.
The parameters in (1.1) satisfy the following assumption.
(ℋ) , , , , , , , , .
Therefore, a weak solution of (1.1) is equivalent to a nonzero critical point of .
Regular semilinear elliptic systems have been studied extensively and many conclusions have been established. For example, Alves et al. studied in  an elliptic system and some important conclusions had been obtained. However, the elliptic systems involving the Hardy inequality have seldom been studied and we only find some results in [8–16]. Thus it is necessary for us to investigate the related singular systems deeply. Among the references above, the elliptic systems involving the Hardy inequality and concave-convex nonlinearities had been studied only in . In this paper, only the case of (1.1) involving multiple strongly-coupled critical terms is considered.
Then the main results of this paper can be concluded in the following theorems and the conclusions are new to the best of our knowledge. It can be verified that the intervals in Theorems 1.1 and 1.2 for the parameters , , μ and q are allowable.
Theorem 1.1 Suppose that (ℋ) holds and . Then problem (1.1) has at least one positive solution.
Theorem 1.2 Suppose that (ℋ) holds, , and . Then there exists such that problem (1.1) has at least two positive solutions for all and satisfying .
This paper is organized as follows. Some preliminary results and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1 and 1.2 are proved in Section 4.
2 The local Palais-Smale condition
denotes the first eigenvalue of the operator L, means the norm of the space , is the dual space of E. for all and . is said to be nonnegative in Ω if and in Ω. is said to be positive in Ω if and in Ω. is a ball in . denotes a quantity satisfying , means as and is a generic infinitesimal value. In particular, the quantity means that there exist the constants such that as ε is small. We always denote positive constants as C and omit dx in integrals for convenience.
Thus, the proof is complete. □
Lemma 2.2 If is a (PS) c -sequence of the functional J, then is bounded in E.
Proof See Hsu [, Lemma 2.2]. □
which is a contradiction. Therefore, the proof of Lemma 2.3 is complete. □
3 Nehari manifold
Lemma 3.1 The functional J is coercive and bounded below on .
Thus, J is coercive and bounded below on . □
Lemma 3.2 Suppose that is a local minimizer of J on and . Then in .
Proof The proof is similar to that of  and the details are omitted. □
Lemma 3.3 for all .
which is a contradiction. □
for all .
There exists a positive constant depending on , , q, N, , and such that for all .
- (ii)Suppose that and . By (1.7), (3.1) and (3.5) we have that
where is a positive constant. □
Proof The proof is similar to that of  and is omitted. □
Then we have the following lemma.
Proof The proof is almost the same as that in [, Lemma 2.7] and is omitted here. □
4 Proof of Theorems 1.1 and 1.2
If , then the functional J has a (PS) -sequence .
If , then the functional J has a -sequence .
Proof The proof is similar to that of  and is omitted. □
Lemma 4.2 Suppose that . Then J has a minimizer such that is a positive solution of (1.1) and .
Therefore, is a nontrivial solution of (1.1).
which is a contradiction. Since and , by Lemma 3.2 we may assume that is a nontrivial nonnegative solution of (1.1).
which is a contradiction.
Finally, from the maximum principle  we deduce that in Ω and is thus a positive solution of (1.1). □
The following results are already known.
Lemma 4.3 
Lemma 4.4 
Suppose that (ℋ) holds, is defined as in (1.6) and are the minimizers of defined as in (1.4). Then and has the minimizers , where .
In particular, for all .
Note that and as t is closed to 0. Thus, is attained at some finite with . Furthermore, , where and are the positive constants independent of ε.
where we have used the assumption .
The proof is thus complete by taking . □
Lemma 4.6 Set . Then for all , problem (1.1) has a positive solution such that and .
This implies that and . From the strong maximum principle  it follows that is a positive solution of (1.1). □
Proof of Theorems 1.1 and 1.2. By Lemma 4.2, we obtain that (1.1) has a positive solution for all . On the other hand, from Lemma 4.6, we can get the second positive solution for all . Since , this implies that and are distinct. □
The author was grateful for the referee’s helpful suggestions and comments.
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