Symmetry and non-existence of solutions to an integral system

In this paper, we consider the nonnegative solutions of the following system of integral form:

$\left\{ \begin{matrix} {{u}_{i}}(x)=\int_{{{\mathbf{R}}^{n}}}{\frac{1}{|x-y{{|}^{n-\alpha }}}}{{f}_{i}}(u(y))dy,\ \ x\in {{\mathbf{R}}^{n}},\ \ i=1,\cdots ,m, \\ 0<\alpha <n,\text{and }\ u(x)=({{u}_{1}}(x),{{u}_{2}}(x),\cdots ,{{u}_{m}}(x)). \\\end{matrix} \right.\ \ \ \ \ \ \left( 1 \right)$

Here $ f_i(u)∈ C^1(\mathbf{R^m_+})\bigcap$$ C^0(\mathbf{\overline{R^m_+}})(i = 1,2,···,m)$ are real-valued functions, nonnegative, homogeneous of degree $ β_{i}$, where $ 0<β_{i} ≤q \frac{n+α}{n-α}$, and monotone nondecreasing with respect to the variables $ u_1, u_2, ···, u_m$. We show that the nonnegative solution $ u = (u_1,u_2,···,u_m)$ is radially symmetric in the critical and subcritical case by method of moving planes in an integral form and $ u$ must be zero in the subcritical case.

Futhermore, we consider the form of $ f_i(u) = \sum_{r = 1}^{k}f_{ir}(u),$ where $ f_{ir}(u)$ are real-valued homogeneous functions of various degrees $ β_{ir}, r = 1,2,···,k$ and $ 0 <β_{ir} ≤q \frac{n+α}{n-α}$. We also show that the radial symmetry property of the nonnegative solution. Due to the homogeneous of degree can be different, the more intricate method is needed to deal with this difficulty.