Asymptotic behavior for solutions of some integral equations

Yutian Lei; Chao Ma

doi:10.3934/cpaa.2011.10.193

Article Contents

2011, Volume 10, Issue 1: 193-207. Doi: 10.3934/cpaa.2011.10.193

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Asymptotic behavior for solutions of some integral equations

Yutian Lei^1, and
Chao Ma^2,

1.
School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097
2.
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309

Received: December 31, 2009

Revised: May 31, 2010

Published: December 2010

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Abstract

In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of the Hardy-Littlewood-Sobolev type in $R^n$

$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,

$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $

We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.

Keywords:

Integral equations,
asymptotic analysis,
singularities,
weighted Hardy-Littlewood-Sobolev inequality.

Mathematics Subject Classification: Primary: 45E10, 45G05.

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