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Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

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

Published online by Cambridge University Press:  13 December 2021

Yuxia Guo  and
Shaolong Peng
Yuxia Guo
Affiliation:
Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China (yguo@mail.tsinghua.edu.cn, psl20@mails.tsinghua.edu.cn)
Shaolong Peng
Affiliation:
Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China (yguo@mail.tsinghua.edu.cn, psl20@mails.tsinghua.edu.cn)
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Abstract

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system:

\[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \]
where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$, $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$, respectively.

Keywords

Pseudo-relativistic Schrödinger systemdirect methods of moving planesLiouville-type theoremsmonotonicitysymmetry

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 35B06: Symmetries, invariants, etc. 35B53: Liouville theorems, Phragmén-Lindelöf theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 1 , February 2023 , pp. 196 - 228
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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