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-Hénon system $$\{\begin{array}{rl}& (-\Delta {)}^{s_1}u(x)=|x{|}^{\alpha }{v}^p(x),x\in {\mathbb{R}}_{+}^n,\\ & (-\Delta {)}^{s_2}v(x)=|x{|}^{\beta }{u}^q(x),x\in {\mathbb{R}}_{+}^n,\\ & u(x)\ge 0,v(x)\ge 0,x\in {\mathbb{R}}_{+}^n,\end{array}$$ 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 $\alpha ,\beta$.

22 pages