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Existence result for semilinear elliptic systems involving critical exponents

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Department of Basic Sciences, Babol University of Technology, Babol, 47148-71167, Iran

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Boundary Value Problems 2012, 2012:119  doi:10.1186/1687-2770-2012-119

 Received: 14 July 2012 Accepted: 4 October 2012 Published: 24 October 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1 Introduction

We consider the following elliptic system:

(1.1)

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].

Before stating the main result, we clarify some terminology. Since our method is variational in nature, we need to define the energy functional of (1.1) on

Then belongs to . A pair of functions is said to be a solution of (1.1) if , and for all , we have

Standard elliptic arguments show that

The following assumptions are needed:

() , and , , , , ,

() , where is the first eigenvalue of L, , are the eigenvalues of the matrix .

The quadratic from is positively defined and satisfies

(1.2)

Our main results are as follows.

Theorem 1.1Suppose () holds. Then for any solutionof problem (1.1), there exists a positive constantsuch that

whereand.

Theorem 1.2Suppose () holds. Then for any positive solutionof problem (1.1), there exists a positive constantsuch thatand

where.

Theorem 1.3Suppose (), () hold. Then the problem (1.1) has a positive solution.

2 Preliminaries

On , we use the norm

Using the Young inequality, the following best constant is well defined:

(2.1)

where is the completion of with respect to the norm .

We infer that is attained in by the functions

where

For all , , , , by the Young and Hardy-Sobolev inequalities, the following constant is well defined on :

(2.2)

Set

where , , satisfies and , , for all small. Then for any , by [9] we have the following estimates:

and for any ,

where , is the volume of the unit ball in .

3 Asymptotic behavior of solutions

Proof of Theorem 1.1 Suppose is a nontrivial solution to problem (1.1). For all define

It is not difficult to verify that and satisfy

(3.1)

Let small enough such that and for . Also, let be a cut-off function. Set

where . Multiplying the first equation of (3.1) by and the second one by respectively and integrating, we have

Note that .

Then

By the Cauchy inequality and the Young inequality, we get

(3.2)

The same result holds for .

By letting , , we have

(3.3)

Using Caffarelli-Kohn-Nirenberg inequality [10], we infer that

(3.4)

Define

Then . Now, from the Hölder inequality, we deduce that

(3.5)

(3.6)

In the sequel, we have

(3.7)

By the choice of , we obtain

(3.8)

So, from (3.4) to (3.8) it follows that

(3.9)

Take and to be a constant near the zero. Letting , we infer that and so

(3.10)

Suppose is sufficiently small such that and is a cut-off function with the properties and in .

Set , .

Then we have the following results:

(3.11)

where we used the Hölder inequality. From (3.9) in combination with (3.11), it follows that

(3.12)

where .

Denote , and , , where , and . Using (3.12) recursively, we get

we have as . Note that the infinite sums on the right-hand side converge, then we obtain that , particularly, we have . Thus,

where .

where . The proof is complete. □

Proof of Theorem 1.2 Suppose is a positive solution to problem (1.1). For all , set

Then

(3.13)

Choose and define for . Let

It is easy to verify that

(3.14)

Combining (3.13) with (3.14), we get

Therefore, by the maximum principle in , we obtain

Thus, for all ,

Taking , we conclude for all .

Similar result also holds for . Therefore, we have

For any ,

For any . This proves the theorem. □

4 Local -condition and the existence of positive solutions

We first establish a compactness result.

Lemma 4.1Suppose that () holds. ThenJsatisfies the-condition for all

Proof Suppose that satisfies and . The standard argument shows that is bounded in .

For some , we have

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 [10], we have

(4.1)

We claim that is finite, and for any , or .

In fact, let be small enough for any , and for , . Let be a smooth cut-off function centered at such that , for , for and . Then

Then we have

By the Sobolev inequality, ; and then we deduce that or , which implies that is finite.

Now, we consider the possibility of concentration at points (), for small enough that for all and for and , . Let be a smooth cut-off function centered at such that , for and . Then

Thus, we have

(4.2)

From (4.1) and (4.2) we derive that , , and then either or . On the other hand, from the above arguments, we conclude that

If for all and , then , which contradicts the assumption that . On the other hand, if there exists an such that or there exists a with , then we infer that

which contradicts our assumptions. Hence, , as in . □

First, under the assumptions (), (), we have the following notations:

where is a minimal point of , and therefore a root of the equation

Lemma 4.2Suppose that () holds. Then we have

(i)

(ii) has the minimizers, , whereare the extremal functions ofdefined as in (2.2).

Proof The argument is similar to that of [6]. □

Lemma 4.3Under the assumptions of (), we have

Proof Suppose () holds. Define the function

Note that and as t is close to 0. Thus, is attained at some finite with . Furthermore, , where and are the positive constants independent of ε. By using (1.2), we have

Note that

(4.3)

and and so .

From (4.3), Lemma 4.2 and Lemma 4.3, it follows that

so . Hence, , and

(4.4)

□

Proof of Theorem 1.3 Set , where

Suppose that () holds. For all , from the Young and Hardy-Sobolev inequalities, it follows that

and there exists a constant small such that

Since as , there exists such that and . By the mountain-pass theorem [14], there exists a sequence such that and , as .

From Lemma 4.2 it follows that

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. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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.

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