Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations

Wenxiong Chen; Chao Jin; Congming Li; Jisun Lim

doi:10.3934/proc.2005.2005.164

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2005, Volume 2005: 164-172. Doi: 10.3934/proc.2005.2005.164 open access

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Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations

1.
Department of Mathematics, Yeshiva University, New York, NY 10033
2.
Department of Applied Mathematics, University of Colorado at Boulder
3.
Department of Applied Mathematics, University of Colorado at Boulder, Campus Box 526, Boulder, CO 80309-0526

Received: August 31, 2004

Revised: January 31, 2005

Published: August 2005

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Abstract

In this paper, we consider systems of integral equations related to the weighted Hardy-Littlewood-Sobolev inequality. We present the symmetry, monotonity, and regularity of the solutions. In particular, we obtain the optimal integrability of the solutions to a class of such systems. We also present a simple method for the study of regularity, which has been extensively used in various forms. The version we present here contains some new developments. It is much more general and very easy to use. We believe the method will be helpful to both experts and non-experts in the field.

Keywords:

Weighted Hardy-Littlewood-Sobolev inequalities,
integral equations and systems,
radial symmetry,
monotonicity,
moving planes in integral forms.

Mathematics Subject Classification: 35J99, 45E10, 45G05.

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