Symmetry and Nonexistence of Positive Solutions for Weighted HLS System of Integral Equations on a Half Space

Abstract

We consider system of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality in a half space. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system is in both supercritical and subcritical cases under some integrability conditions. Ruling out these nonexistence results, we also discuss the positive solutions of the integral system in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric about -axis, which is much more general than the main result of Zhuo and Li, 2011.

1. Introduction

In [1], Jin and Li studied the weighted HLS system of nonlinear equations in : where and .

By the method of moving planes in integral forms they derived symmetry and monotonicity of positive solutions of (1) under some integrability conditions.

Theorem 1 (see [1]). Let the pair be a positive solution of system (1) with , and , , and . Then and are radially symmetric and decreasing about some point .

Jin and Li [2] and Chen et al. [3] also discussed the regularity of solutions to (1).

Let be the upper half Euclidean space In this paper, we want to consider the similar integral system in the half space as (1). More precisely, we discuss the following weighted HLS type system of nonlinear equations in : where , , , , , , and here is the reflection point of about the plane .

Similar to some integral systems or PDEs systems, the integral system (3) is usually divided into three cases according to the value of exponents . We say that system (3) is in critical case when the pair satisfies the relation It is in supercritical case when “” holds; and in subcritical case when “” holds; that is In the special case, where and , system (3) reduces to and system (7) is closely related to the following system of PDEs with Navier boundary conditions: In particular, when is an even number, the authors ([4]) proved the equivalence between the two systems (7) and (8) under some mild growth condition.

Symmetry of solutions to integral system (8) was established by Zhuo and Li [5]. They proved that in critical case , any pair of positive solutions of (7) with and is rotationally symmetric about some line parallel to -axis. Under the same integrability conditions, in [6], we obtained the nonexistence of positive solutions of (7).

The general case is that, for and in (3), there are few results concerning symmetry and nonexistence for this doubled weighted system. In this paper, by the Pohozaev type identity in integral form, we present a Liouville type theorem when the system (3) is in both supercritical and subcritical cases under some integrability conditions. Based on these nonexistence results, we discuss the positive solutions of (3) in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric about -axis. To carry on the moving of planes, we explore global features of the integral equations and estimate certain integral norms. This is the essence of the method of moving planes in integral forms. The readers who are interested in the integral system and the applications of this method may consult [7–10] and the references therein.

The paper is organized as follows.

In Section 2, by the Pohozaev type identity in integral forms, we prove the following nonexistence results.

Theorem 2. Suppose that are nonnegative solutions of (3) with , .(i)If and are both supercritical, that is, or(ii)if and are both subcritical, that is, then and .

Based on these results and ruling out cases where there are no solutions, we are only interested in critical case (5). In Section 3, by means of method of moving planes in integral form, we establish rotational symmetry of solutions of (3) in critical case (5) as follows.

Theorem 3. Assume that , and satisfy (5). If is a pair of positive solutions of (3), then is rotationally symmetric about axis.

Remark 4. When , Theorem 3 is coincident with the result in [5].

2. Proof of Theorem 2

In this section we will prove the nonexistence of positive solutions to the weighted HLS type system (3). These nonexistence results, known as Liouville type theorems, are useful in deriving existence, a priori estimate, regularity, and asymptotic analysis of solutions.

A celebrated result of S. I. Pohozaev is known as the Pohozaev identity. This classical result has many consequences, the most immediate one being the nonexistence of nontrivial bounded solutions to PDE. Here we apply the Pohozaev type identity in integral forms to the integral system (3) (see in [9, 11]).

For any , there holds By an elementary calculation,

Noting , differentiating both sides of (11) with respect to and letting , we have Let be the upper half ball in the half space in . Multiplying left side of (13) by and integrating on yields Similarly, we also have

Since Thus, there exists a sequence such that Let ; by (14), (15), and (17), we have On the other hand, There also holds Using Combining the fact , (19), and (20), we have By (18) and (22), we have Hence, if or hold, it follows that and .

This completes the proof of Theorem 2.

Remark 5. In [11], the authors consider another weighted HLS type integral system and showed the Liouville type theorem as follows.

Theorem 6 (see [11]). Suppose that are positive solutions of (26) when and are both subcritical; that is and . If , and , , then and .

When in system (26) or in system (3), the two systems reduce to the simple integral system (7). In this special case, we can find that Theorem 6 is coincident with case (ii) in Theorem 2.

3. Proof of Theorem 3

In this section, we will consider rotational symmetry of weighted HLS type system (3) in critical case (5).

Firstly, we need the following weighted HLS inequality.

Lemma 7 (see [12]). Let , , , , and . Then One can also write the weighted HLS inequality in another form. Let Then where , .

For a given real number , define Let be the reflection of the point about the plane . Set

Lemma 8 (see [8, 13]). For , , one has

Lemma 9. Let be any pair of positive solutions of (3) in critical case (5); for any and , one has

Proof. Through the calculation, we have By the assumption , we have
Similarly, we have

Proof of Theorem 3. Step 1. We will show that for sufficiently negative , Define We prove that, for sufficiently negative , both and must be empty and thus (37) holds.
In fact, by Lemma 9 and the mean value theorem, we have, for , where is valued between and ; therefore, on , we have By Lemma 7 and the Hölder inequality, we have where and . It easy to show that . Combining (41) and (42), we arrive The conditions and make us able to choose sufficiently negative , so that Now inequality (43) implies and therefore must be measure zero. Similarly, one can show that is measure zero. Therefore (37) holds.
Step 2. Inequality (37) provides a starting point to move the plane . Now we start from the neighborhood of and move the plane to the right as long as (37) holds to the limiting position. More precisely, define We will prove that . On the contrary, we suppose . We show that and are symmetric about the plane ; that is Otherwise, on , We show that the plane can be moved further to the right. More precisely, there exists an such that, for ,
Without loss of generality, we assume by Lemma 9, we have in fact in the interior of . Let Then obviously has measure zero and . The same argument above is also true for the other solution of (3). From (41) and (42), we deduce Again the conditions that and ensure that one can choose sufficiently small, so that, for all in , The method to verify this inequality is standard and the proofs of the rest are similar to the proof in paper [6, 11, 14].
Now by (52) and (53), we have , and therefore must be measure zero. Similarly, must also be measure zero. Hence, for these values of , we have This (47) must hold and therefore both and are symmetric about the plane .
Now we show that the plane cannot stop before hitting the origin. Otherwise, assume that the plane stops at . By the fact that , we have
This contradicts with (47).
As the direction of can be chosen arbitrarily, we derive that is rotationally symmetric about axis. This completes the proof of Theorem 3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Wenxiong Chen and the referees for their valuable suggestions and comments. This work is supported by Grant (nos. U1304101 and 11171091) of NSFC and NSF of Henan Province (no. 132300410141).