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.
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. □
- Cao D, Han P: Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differ. Equ. 2006, 224: 332-372. 10.1016/j.jde.2005.07.010MathSciNetView ArticleGoogle Scholar
- Hsu TS: Multiple positive solutions for semilinear elliptic equations involving multi-singular inverse square potentials and concave-convex nonlinearities. Nonlinear Anal. 2011, 74: 3703-3715. 10.1016/j.na.2011.03.016MathSciNetView ArticleGoogle Scholar
- Kang D: On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents. Acta Math. Sin. Engl. Ser. 2009, 25: 435-444. 10.1007/s10114-008-6450-7MathSciNetView ArticleGoogle Scholar
- Abdellaoui B, Felli V, Peral I:Some remarks on systems of elliptic equations doubly critical in the whole . Calc. Var. Partial Differ. Equ. 2009, 34: 97-137. 10.1007/s00526-008-0177-2MathSciNetView ArticleGoogle Scholar
- Bouchekif M, Nasri Y: On a singular elliptic system at resonance. Ann. Mat. Pura Appl. 2010, 189: 227-240. 10.1007/s10231-009-0106-9MathSciNetView ArticleGoogle Scholar
- Huang Y, Kang D: On the singular elliptic systems involving multiple critical Sobolev exponents. Nonlinear Anal. 2011, 74: 400-412. 10.1016/j.na.2010.08.051MathSciNetView ArticleGoogle Scholar
- Kang D: Semilinear systems involving multiple critical Hardy-Sobolev exponents and three singular points. Appl. Math. 2011, 218: 4514-4522.Google Scholar
- Kang D, Peng S: Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents. Sci. China Math. 2011, 54(2):243-256. 10.1007/s11425-010-4131-3MathSciNetView ArticleGoogle Scholar
- 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. 2008, 27(3):91-95.Google Scholar
- Caffarelli L, Kohn R, Nirenberg L: First order interpolation inequality with weights. Compos. Math. 1984, 53: 259-275.MathSciNetGoogle Scholar
- Cai, M, Kang, D: Concentration-compactness principles for the systems of critical elliptic equations. Acta Math. Sci. Ser. B Engl. Ed. (to appear)Google Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations: the limit case I. Rev. Mat. Iberoam. 1985, 1: 45-121.View ArticleGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations: the limit case II. Rev. Mat. Iberoam. 1985, 1: 145-201.View ArticleGoogle Scholar
- Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
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