An integral equation on half space

Let $R^n_+$ be the $n$-dimensional upper half Euclidean space, and let $\alpha$ be any real number satisfying $0<\alpha<n.$ In this paper, we consider the integral equation

$u(x)=\int_{R^n_+} (\dfrac{1}{\vert x-y\vert^{n-\alpha}}-\dfrac{1}{\vert x^*-y\vert^{n-\alpha}})u^\tau(y), u(x)>0, \forall x \in R_+^n,$

(1)

where $\tau =\dfrac{n+\alpha}{n-\alpha}$, and $x^*=(x_1,\cdots,x_{n-1},-x_n)$ is the reflection of the point $x$ about the hyperplane $x_n =0$. We use a new type of moving plane method in integral forms introduced by Chen, Li and Ou to establish the regularity and rotational symmetry of the solution of the above integral equation.