An integral equation on half space

Abstract:

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 \begin{equation} u(x)=\int _{R^n_+} (\dfrac {1}{|x-y|^{n-\alpha }}-\dfrac {1}{|x^*-y|^{n-\alpha }})u^\tau (y), u(x)>0, \forall x \in R_+^n, \end{equation} 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.