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Nonexistence of anti-symmetric solutions for fractional Hardy–Hénon system
Published online by Cambridge University Press: 15 May 2023
Abstract
We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$
for the following fractional Hardy–Hénon system:\[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), & x\in\mathbb{R}_+^n, \\ u(x)\geq 0, & v(x)\geq 0,\ x\in\mathbb{R}_+^n, \end{array}\right. \]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 at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropriate condition of $\alpha, \beta$
.
Keywords
MSC classification
Primary:
35A01: Existence problems
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 3 , June 2024 , pp. 862 - 886
- Copyright
- Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
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