Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$

Let $n, m$ be a positive integer and let $R_+^n$ be the$n$-dimensional upper half Euclidean space. In this paper, we studythe following integral equation\begin{eqnarray}u(x)=\int_{R_+^n}G(x,y)u^pdy,\end{eqnarray}where\begin{eqnarray*}G(x,y)=\frac{C_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_n y_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{n/2}}dz,\end{eqnarray*}$C_{n}$ is a positive constant, $0 1$. Using the method ofmoving planes in integral forms, we show that equation (1)has no positive solution.