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:
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
Our main results are as follows.
Theorem 1.1Suppose ( ) holds. Then for any solution of problem (1.1), there exists a positive constant such that
where and .
Theorem 1.2Suppose ( ) holds. Then for any positive solution of problem (1.1), there exists a positive constant such that and
Theorem 1.3Suppose ( ), ( ) hold. Then the problem (1.1) has a positive solution.
On , we use the norm
Using the Young inequality, the following best constant is well defined:
where is the completion of with respect to the norm .
We infer that is attained in by the functions
For all , , , , by the Young and Hardy-Sobolev inequalities, the following constant is well defined on :
where , , satisfies and , , for all small. Then for any , by  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
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 .
By the Cauchy inequality and the Young inequality, we get
The same result holds for .
By letting , , we have
Using Caffarelli-Kohn-Nirenberg inequality , we infer that
Then . Now, from the Hölder inequality, we deduce that
In the sequel, we have
By the choice of , we obtain
So, from (3.4) to (3.8) it follows that
Take and to be a constant near the zero. Letting , we infer that and so
Suppose is sufficiently small such that and is a cut-off function with the properties and in .
Set , .
Then we have the following results:
where we used the Hölder inequality. From (3.9) in combination with (3.11), it follows that
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 . The proof is complete. □
Proof of Theorem 1.2 Suppose is a positive solution to problem (1.1). For all , set
Choose and define for . Let
It is easy to verify that
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 , we have
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
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
(ii) has the minimizers , , where are the extremal functions of defined as in (2.2).
Proof The argument is similar to that of . □
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
and and so .
From (4.3), Lemma 4.2 and Lemma 4.3, it follows that
so . Hence, , and
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 , 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. □
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.
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