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November  2009, 8(6): 1925-1932. doi: 10.3934/cpaa.2009.8.1925

Symmetry and monotonicity for a system of integral equations

Changlu Liu 1, and  Shuangli Qiao 1,

1. 

Department of Mathematical Sciences, Henan Normal University, Xinxiang 453007, China, China

Received  October 2008 Revised  April 2009 Published  August 2009

In this paper, we consider radial symmetry of positive solutions for a system of three integral equations in $R^n$. Under some mild integrability conditions, we prove that all the solutions are radially symmetric and monotone decreasing about some point. This generalizes a recent result of Chen, Li, and Ou [4]. To establish the symmetry, we use an integral form of the method of moving planes which is quite different from the traditional method of moving planes for PDEs. We also generalize our result to a system containing any number of integral equations.
Keywords: Hardy-Littlewood-Sobolev inequalities, radial symmetry, moving planes in integral forms., systems of integral equations.
Mathematics Subject Classification: 35J99, 45E1.
Citation: Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
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