An integral equation involving Bessel potentials on half space

Abstract

In this article we consider the following integral equation involving Bessel potentials on a half space $\mathbb{R}^n_+ $:\begin{eqnarray}u(x)=\int_{ \mathbb{R}^n_+ }\{g_\alpha(x-y)-g_\alpha(\bar x-y)\} u^\beta(y) dy,\;\;x\in \mathbb{R}^n_+,\end{eqnarray}where $\alpha>0$, $\beta>1$, $\barx$ is the reflection of $x$about $x_n=0$, and $g_\alpha(x)$ denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positivesolution for the integral equation.