Symmetry and nonexistence of positive solutions for fractional systems

Abstract

This paper is devoted to study the nonexistence results of positive solutions for the following fractional H$\acute{e}$non system \begin{eqnarray*}\left\{ \begin{array}{lll} &(-\triangle)^{α/2}u=|x|^av^p,~~~&x\in R^n, &(-\triangle)^{α/2}v=|x|^bu^q,~~~ &x\in R^n, &u\geq0, v\geq 0, \end{array} \right. \end{eqnarray*} where $0<\alpha <2$, $0<p,q<\infty$, $a$, $b$ $\ge 0$, $n\ge 2$. Using a direct method of moving planes, we prove non-existence of positive solution in the subcritical case.