Qualitative properties of solutions for an integral equation

Wenxiong Chen; Congming Li; Biao Ou

doi:10.3934/dcds.2005.12.347

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2005, Volume 12, Issue 2: 347-354. Doi: 10.3934/dcds.2005.12.347

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Qualitative properties of solutions for an integral equation

1.
Department of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033
2.
Department of Applied Mathematics, University of Colorado at Boulder
3.
Department of Mathematics, University of Toledo, Toledo OH 43606

Received: August 2003

Revised: June 2004

Published: February 2005

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Abstract

Let $n$ be a positive integer and let $ 0 < \alpha < n.$ In this paper, we study more general integral equation

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

We establish regularity, radial symmetry, and monotonicity of the solutions. We also consider subcritical cases, super critical cases, and singular solutions in all cases; and obtain qualitative properties for these solutions.

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

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