Symmetry and nonexistence of positive solutions for fractional systems
  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)^{\alpha/2}u=|x|^av^p,~~~&x\in R^n,   &(-\triangle)^{\alpha/2}v=|x|^bu^q,~~~ &x\in R^n,   &u\geq0, v\geq 0,   \end{array}   \right. \end{eqnarray*} where $0<\alpha<2$, $0<p,q<\infty$, $a$, $b$ $\geq0$, $n\geq2$. Using a direct method of moving planes, we prove non-existence of positive solution in the subcritical case. 
Pei Ma, Yan Li, Jihui Zhang
math.AP