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February  2005, 12(2): 347-354. doi: 10.3934/dcds.2005.12.347

Qualitative properties of solutions for an integral equation

Wenxiong Chen 1, , Congming Li 2, and  Biao Ou 3,

1. 

Department of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033, United States
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  December 2004

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.

Keywords: moving planes., Kelvin type transforms, radial symmetry, monotonicity, Regularity, subcritical and super critical cases, singular solutions.
Mathematics Subject Classification: 35J99, 45E10, 45G0.
Citation: Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347
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