Qualitative properties of solutions for an integral equation

Abstract

Let $n$ be a positive integer and let $ 0 < \alpha < n.$ In this paper, we study more general integral equation <p align="center"> $ u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy. <p align="left" class="times"> 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.