Symmetry and monotonicity for a system of integral equations

Changlu Liu; Shuangli Qiao

doi:10.3934/cpaa.2009.8.1925

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2009, Volume 8, Issue 6: 1925-1932. Doi: 10.3934/cpaa.2009.8.1925

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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

Received: October 2008

Revised: April 2009

Published: November 2009

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Abstract

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.

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Mathematics Subject Classification: 35J99, 45E10.

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