Nonexistence of anti-symmetric solutions for fractional Hardy-Hénon System

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