Radial symmetry of nonnegative solutions for nonlinear integral systems

In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system\begin{document}$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0<\alpha<n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $\end{document}where \begin{document}$ 0<a_i/2<\alpha $\end{document} , \begin{document}$ f_i(u) $\end{document} , \begin{document}$ 1\leq i\leq m $\end{document} , are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables \begin{document}$ u_1 $\end{document} , \begin{document}$ u_2 $\end{document} , \begin{document}$ \cdots $\end{document} , \begin{document}$ u_m $\end{document} . By the method of moving planes in integral forms, we show that the nonnegative solution \begin{document}$ u = (u_1,u_2,\cdots,u_m) $\end{document} is radially symmetric when \begin{document}$ f_i $\end{document} satisfies some monotonicity condition.