Nonexistence of anti-symmetric solutions for fractional Hardy-H\'{e}non System

TL;DR

This paper proves the nonexistence of anti-symmetric solutions for a fractional Hardy-Hénon system in certain parameter regimes using moving sphere and plane methods, especially when the fractional orders are equal.

Contribution

It establishes nonexistence results for anti-symmetric solutions of the fractional Hardy-Hénon system under specific conditions, extending previous knowledge to fractional orders and particular parameter domains.

Findings

Nonexistence of anti-symmetric solutions in certain parameter regions.

Results apply to cases where fractional orders are equal, above the fractional Sobolev hyperbola.

Uses moving sphere and moving plane methods for proofs.

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 $\alpha, \beta$.