Moving sphere approach to a general weighted integral equation

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 $$u(x)={\int }_{{\mathbf{R}}^n}\mathrm{\mid }x-y{\mathrm{\mid }}^{p}f(\mathrm{\mid }y\mathrm{\mid },u(y))dy\text{in }{\mathbf{R}}^n\setminus \{\mathbf{\text{0}}\}\mathrm{.}$$ 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.