Existence result for semilinear elliptic systems involving critical exponents
© Khademloo et al.; licensee Springer 2012
Received: 14 July 2012
Accepted: 4 October 2012
Published: 24 October 2012
In this paper we deal with the existence of a positive solution for a class of semilinear systems of multi-singular elliptic equations which involve Sobolev critical exponents. In fact, by the analytic techniques and variational methods, we prove that there exists at least one positive solution for the system.
MSC: 35J60, 35B33.
Keywordssemilinear elliptic system nontrivial solution critical exponent variational method
where () is a smooth bounded domain such that , , , are different points, , , , , , , .
We work in the product space , where the space is the completion of with respect to the norm .
In resent years many publications [1–3] concerning semilinear elliptic equations involving singular points and the critical Sobolev exponent have appeared. Particularly in the last decade or so, many authors used the variational method and analytic techniques to study the existence of positive solutions of systems of the form of (1.1) or its variations; see, for example, [4–8].
The following assumptions are needed:
() , and , , , , ,
() , where is the first eigenvalue of L, , are the eigenvalues of the matrix .
Our main results are as follows.
where and .
Theorem 1.3 Suppose (), () hold. Then the problem (1.1) has a positive solution.
where is the completion of with respect to the norm .
where , is the volume of the unit ball in .
3 Asymptotic behavior of solutions
Note that .
The same result holds for .
Suppose is sufficiently small such that and is a cut-off function with the properties and in .
Set , .
where . The proof is complete. □
Taking , we conclude for all .
For any . This proves the theorem. □
4 Local -condition and the existence of positive solutions
We first establish a compactness result.
Proof Suppose that satisfies and . The standard argument shows that is bounded in .
We claim that is finite, and for any , or .
By the Sobolev inequality, ; and then we deduce that or , which implies that is finite.
which contradicts our assumptions. Hence, , as in . □
has the minimizers , , where are the extremal functions of defined as in (2.2).
Proof The argument is similar to that of . □
and and so .
Since as , there exists such that and . By the mountain-pass theorem , there exists a sequence such that and , as .
By Lemma 4.1 there exists a subsequence of , still denoted by , such that strongly in . Thus, we get a critical point of J satisfying (1.1), and c is a critical value. Set .
Replacing respectively u, ν with and in terms of the right-hand side of (1.1) and repeating the above process, we can get a nonnegative nontrivial solution of (1.1). If , we get by (1.1) and the assumption . Similarly, if , we also have . There, . From the maximum principle, it follows that in Ω. □
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