Symmetry and Nonexistence of Positive Solutions for Weighted HLS System of Integral Equations on a Half Space

Abstract

We consider system of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality in a half space. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system is in both supercritical and subcritical cases under some integrability conditions. Ruling out these nonexistence results, we also discuss the positive solutions of the integral system in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric about $x_n$-axis, which is much more general than the main result of Zhuo and Li, 2011.