Moving sphere approach to a general weighted integral equation

Abstract

Let $p$ be positive and $n\ge 3$ be an integer. Let $f(\cdot ,\cdot ):{\mathbf{R}}_{+}\times {\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ be a continuous function. In this paper, we are concerned with positive solutions to the following integral equation \[ u(x)= \int_{\mathbf{R}^n} |x-y|^p f(|y|,u(y)) dy \quad\text{in }\mathbf{R}^n\setminus\{\textbf{0}\}. \] By imposing some suitable conditions on $f$, we obtain the radially symmetry property of positive solutions to the above equation by using the method of moving spheres in integral form.