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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 504282, 5 pages
Necessary Conditions for Existence Results of Some Integral System
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.
In this paper, we give some necessary conditions for the existence of positive solutions for integral systems.
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 , the authors studied the symmetric property and the uniqueness of solutions for problem (2) in . Later, they studied the integral system (1) in . Also, after the work of , Li studied the general form of (2) in . For the case , he obtained similar results to  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 , 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 , 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 .
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 .
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  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 , 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.
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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