Qualitative properties of stable solutions to some supercritical problems

Yong Liu; Kelei Wang; Juncheng Wei; Ke Wu; Yong Liu; Kelei Wang; Juncheng Wei; Ke Wu

doi:10.3934/era.2022084

Electronic Research Archive

2022, Volume 30, Issue 5: 1668-1690. doi: 10.3934/era.2022084

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Qualitative properties of stable solutions to some supercritical problems

1.
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui Province, 230026, China
2.
School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei Province, 430072, China
3.
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada
4.
School of Mathematics and Statistics, Xian Jiaotong University, Xian, Shanxi Province, 710049, China

Received: 06 July 2021 Revised: 14 February 2022 Accepted: 15 February 2022 Published: 24 March 2022

In this paper, we study symmetry properties of stable solutions to the Lane-Emden equation

$ \Delta u+|u|^{p-1}u = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $

with $ n\geq 11 $, $ p $ in a suitable range and the Liouville equation

$ \Delta u+e^{u} = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $

with $ n = 10 $.
- Lane-Emden equation,
- Liouville equation,
- stable solution,
- rigidity results,
- symmetry
Citation: Yong Liu, Kelei Wang, Juncheng Wei, Ke Wu. Qualitative properties of stable solutions to some supercritical problems[J]. Electronic Research Archive, 2022, 30(5): 1668-1690. doi: 10.3934/era.2022084

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Abstract

In this paper, we study symmetry properties of stable solutions to the Lane-Emden equation

$ \Delta u+|u|^{p-1}u = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $

with $ n\geq 11 $, $ p $ in a suitable range and the Liouville equation

$ \Delta u+e^{u} = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $

with $ n = 10 $.

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