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.
Keywords:semilinear elliptic system; nontrivial solution; critical exponent; variational method
We consider the following elliptic system:
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].
Standard elliptic arguments show that
The following assumptions are needed:
Our main results are as follows.
Using the Young inequality, the following best constant is well defined:
where , , satisfies and , , for all small. Then for any , by  we have the following estimates:
3 Asymptotic behavior of solutions
By the Cauchy inequality and the Young inequality, we get
Using Caffarelli-Kohn-Nirenberg inequality , we infer that
In the sequel, we have
So, from (3.4) to (3.8) it follows that
Then we have the following results:
where we used the Hölder inequality. From (3.9) in combination with (3.11), it follows that
It is easy to verify that
Combining (3.13) with (3.14), we get
We first establish a compactness result.
Therefore, is a solution to (1.1). Then by the concentration-compactness principle [11-13] and up to a subsequence, there exist an at most countable set , a set of different points , nonnegative real numbers , , , and , , () such that the following convergence holds in the sense of measures:
By the Sobolev inequalities , we have
Then we have
Thus, we have
Proof The argument is similar to that of . □
From (4.3), Lemma 4.2 and Lemma 4.3, it follows that
Since as , there exists such that and . By the mountain-pass theorem , there exists a sequence such that and , as .
From Lemma 4.2 it follows that
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. □
The authors declare that they have no competing interests.
Each of the authors, SK, MF and OKK contributed to each part of this work equally and read and approved the final version of the manuscript.
Cao, D, Han, P: Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differ. Equ.. 224, 332–372 (2006). Publisher Full Text
Hsu, TS: Multiple positive solutions for semilinear elliptic equations involving multi-singular inverse square potentials and concave-convex nonlinearities. Nonlinear Anal.. 74, 3703–3715 (2011). Publisher Full Text
Kang, D: On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents. Acta Math. Sin. Engl. Ser.. 25, 435–444 (2009). Publisher Full Text
Abdellaoui, B, Felli, V, Peral, I: Some remarks on systems of elliptic equations doubly critical in the whole . Calc. Var. Partial Differ. Equ.. 34, 97–137 (2009). Publisher Full Text
Bouchekif, M, Nasri, Y: On a singular elliptic system at resonance. Ann. Mat. Pura Appl.. 189, 227–240 (2010). Publisher Full Text
Huang, Y, Kang, D: On the singular elliptic systems involving multiple critical Sobolev exponents. Nonlinear Anal.. 74, 400–412 (2011). Publisher Full Text
Kang, D, Peng, S: Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents. Sci. China Math.. 54(2), 243–256 (2011). Publisher Full Text
Kang, D, Huang, Y, Liu, S: Asymptotic estimates on the extremal functions of a quasi-linear elliptic problem. J. South-Central Univ. Natl. Nat. Sci. Ed.. 27(3), 91–95 (2008). PubMed Abstract | Publisher Full Text
Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text