Nonexistence of anti-symmetric solutions for fractional Hardy–Hénon system

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ for following fractional Hardy–Hénon system: \[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), u(x)\geq 0, v(x)\geq 0,\ \end{array}\right. \] where $0< s_1,s_2<1$ , $n>2\max \{s_1,s_2\}$ . Nonexistence of are obtained in some appropriate domains $(p,q)$ under corresponding assumptions $\alpha,\beta$ via methods moving spheres and planes. Particularly, case $s_1=s_2$ one our results shows that domain nonexistence with decay conditions at infinity hold true, locates above Sobolev's hyperbola condition $\alpha, \beta$