Symmetry and nonexistence results for a fractional Hénon-Hardy system on a half-space

Abstract

We study the fractional Henon-Hardy system \begin{aligned}\begin{cases}(-\Delta )^{s/2} u(x) = |x|^\alpha v^p(x), & x\in \mathbb{R}^n_+, \\(-\Delta )^{s/2} v(x) = |x|^\beta u^q(x), & x\in \mathbb{R}^n_+, \\ u(x)=v(x)=0, & x\in \mathbb{R}^n\setminus \mathbb{R}^n_+,\end{cases}\end{aligned} where $n\ge 2$, $0\lt s\lt 2$, $\alpha ,\beta >-s$ and $p,q\ge 1$. We also consider an equivalent integral system. By using a direct method of moving planes, we prove some symmetry and nonexistence results for positive solutions under various assumptions on $\alpha $, $\beta $, $p$ and $q$.