# American Institute of Mathematical Sciences

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

## Qualitative properties of solutions for an integral equation

 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. Citation: Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347  [1] Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022 [2] Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 [3] Alfonso Castro, Shu-Zhi Song. 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