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`Abstract and Applied AnalysisVolume 2013 (2013), Article ID 504282, 5 pageshttp://dx.doi.org/10.1155/2013/504282`
Research Article

## 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

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 [718], 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.

#### References

1. E. H. Lieb, “Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,” Annals of Mathematics. Second Series, vol. 118, no. 2, pp. 349–374, 1983.
2. W. Chen, C. Li, and B. Ou, “Classification of solutions for an integral equation,” Communications on Pure and Applied Mathematics, vol. 59, no. 3, pp. 330–343, 2006.
3. W. Chen, C. Li, and B. Ou, “Classification of solutions for a system of integral equations,” Communications in Partial Differential Equations, vol. 30, no. 1–3, pp. 59–65, 2005.
4. Y. Y. Li, “Remark on some conformally invariant integral equations: the method of moving spheres,” Journal of the European Mathematical Society, vol. 6, no. 2, pp. 153–180, 2004.
5. X. Yu, “Liouville type theorems for integral equations and integral systems,” Calculus of Variations and Partial Differential Equations, vol. 46, no. 1-2, pp. 75–95, 2013.
6. X. Yu, “Liouville type theorems for singular integral equations and integral systems,” Calculus of Variations and Partial Differential Equations, vol. 46, no. 1-2, pp. 75–95, 2013.
7. W. Chen and C. Li, “An integral system and the Lane-Emden conjecture,” Discrete and Continuous Dynamical Systems. Series A, vol. 24, no. 4, pp. 1167–1184, 2009.
8. W. Chen and C. Li, “Classification of positive solutions for nonlinear differential and integral systems with critical exponents,” Acta Mathematica Scientia. Series B. English Edition, vol. 29, no. 4, pp. 949–960, 2009.
9. W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, vol. 4 of AIMS Series on Differential Equations & Dynamical Systems, American Institute of Mathematical Sciences, Springfield, Mo, USA, 2010.
10. W. Chen and C. Li, “Radial symmetry of solutions for some integral systems of Wolff type,” Discrete and Continuous Dynamical Systems. Series A, vol. 30, no. 4, pp. 1083–1093, 2011.
11. F. Hang, X. Wang, and X. Yan, “An integral equation in conformal geometry,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 26, no. 1, pp. 1–21, 2009.
12. C. Jin and C. Li, “Symmetry of solutions to some systems of integral equations,” Proceedings of the American Mathematical Society, vol. 134, no. 6, pp. 1661–1670, 2006.
13. L. Ma and D. Chen, “A Liouville type theorem for an integral system,” Communications on Pure and Applied Analysis, vol. 5, no. 4, pp. 855–859, 2006.
14. L. Ma and D. Chen, “Radial symmetry and monotonicity for an integral equation,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 943–949, 2008.
15. L. Ma and D. Chen, “Radial symmetry and uniqueness for positive solutions of a Schrödinger type system,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 379–385, 2009.
16. C. Ma, W. Chen, and C. Li, “Regularity of solutions for an integral system of Wolff type,” Advances in Mathematics, vol. 226, no. 3, pp. 2676–2699, 2011.
17. X. Xu, “Exact solutions of nonlinear conformally invariant integral equations in ${R}^{3}$,” Advances in Mathematics, vol. 194, no. 2, pp. 485–503, 2005.
18. X. Xu, “Uniqueness and non-existence theorems for conformally invariant equations,” Journal of Functional Analysis, vol. 222, no. 1, pp. 1–28, 2005.
19. X. Xu, “Uniqueness theorem for integral equations and its application,” Journal of Functional Analysis, vol. 247, no. 1, pp. 95–109, 2007.
20. W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, vol. 120 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1989.