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

Part of: Partial differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  15 May 2023

Jiaqi Hu  and
Zhuoran Du
Jiaqi Hu
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, China ([email protected], [email protected])
Zhuoran Du
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, China ([email protected], [email protected])
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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

Anti-symmetric solutionsHardy–Hénon systemLiouville theoremmethod of moving planesmethod of moving spheres

MSC classification

Primary: 35A01: Existence problems
Secondary: 35R11: Fractional partial differential equations 35B09: Positive solutions 35B53: Liouville theorems, Phragmén-Lindelöf theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 3 , June 2024 , pp. 862 - 886
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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