Abstract and Applied Analysis

Volume 2013 (2013), Article ID 504282, 5 pages

http://dx.doi.org/10.1155/2013/504282

Research Article

Necessary Conditions for Existence Results of Some Integral System

Yongxia Hua¹ and Xiaohui Yu²

¹Center of Applied Math, Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen, Guangdong 518055, China

²Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, China

Received 3 May 2013; Accepted 13 July 2013

Academic Editor: Jaume Giné

Copyright © 2013 Yongxia Hua and Xiaohui Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we give some necessary conditions for the existence of positive solutions for integral systems.

1. Introduction

In this paper, we study the necessary condition for the existence of positive solutions for the following integral system:

where , ,  and   are real parameters.

As for one single equation

there are a lot of results of this problem. If with , then problem (2) is equivalent to the following differential equation:

This problem has been widely studied in the past few years. For example, in order to answer a question raised by Lieb in [1], the authors studied the symmetric property and the uniqueness of solutions for problem (2) in [2]. Later, they studied the integral system (1) in [3]. Also, after the work of [2], Li studied the general form of (2) in [4]. For the case , he obtained similar results to [2] but with less regularity requirement. For the case , he shows that if problem (2) has a nonnegative solution in and , then . The main ingredients in these papers are the moving plane method and moving sphere method based on the maximum principle of integral forms. This method has been widely used in other works. For example, inspired by these works, the author studied the Liouville-type theorems for problems (1) and (2) with general nonlinearities in [5, 6]. For further results of this type of integral equations, see [7–18], and so forth. We note that all these results concern the cases and . A natural question is whether similar results hold for or . We note that the case and is quite different from the case or . Generally speaking, the moving plane method or the moving sphere method does not work in the latter case, so we have to look for other methods. In a recent paper [19], the author give a sufficient and necessary condition for the existence of positive solutions for problem (2) with . Based on some integral estimates, the author proved that problem (2) possesses a positive solution if and only if . Inspired by the work of [19], we first study the integral system (1) with . Our main result is the following theorem.

Theorem 1. Suppose that and problem (1) possesses a positive solution; then

As for , we have the following nonexistence result.

Theorem 2. If , then problem (1) possesses no positive solution provided that or .

This paper is organized as follows. We prove Theorem 1 in Section 2. The proof of Theorem 2 is completed in Section 3.

2. Proof of Theorem 1

We first claim that and .

In fact, we infer from

that . Also, it follows from (1) that

Now taking limit in (6) by letting , we obtain

We point out that we can take the limit under the integral sign because of the dominated convergence theorem. In fact, we note that when and , we have

It is easy to check that .

It follows from (7) that there exist and such that

for . Finally, we have

which further implies that . Similarly, we have .

Next, we can prove as in [19] that

in the sense of distribution. Hence, we infer from (11) that

Now we choose a cut-off function satisfying , ,   for and for . For any , if we multiply (12) by and integrate over , then we get

While the left-hand side of (13) equals

it follows from

that

as by . Thus we conclude that

While the right-hand side of (13) equals

it can be checked as in [19] that

Hence, by letting in (19) we get

We infer from (13), (18), and (21) that

Similarly, we can prove that

The above two equations imply that

On the other hand, since

we have

by taking into account that . Similarly, we have

Then it follows from (26) and (27) that

Finally, we infer from (24) and (28) that

This completes the proof of Theorem 1.

3. Proof of Theorem 2

We assume that without loss of generality. First, we note that by Lemma   in [20], we have, for all ,

Similarly, we have

for any .

If we choose , , and  , then we can infer from the Holder inequality that

That is,

Since , so if , then we have . Multiplying both sides of (33) by , we get

Since , we have . Hence the left-hand side of (34) goes to infinity as , which is a contradiction. This completes the proof.

Acknowledgments

The first author is supported by SZPP (no. KQCX20120802140634893), Guangdong S&T Major Project (no. 2012A080104014), and SRF for ROCS, SEM. Xiaohui Yu is supported by NSFC no. 11101291. The authors would like to thank the anonymous referees for their valuable suggestions.

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