Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus

Ruofei Yao; Yi Li; Hongbin Chen

doi:10.3934/dcds.2018122

Article Contents

2019, Volume 39, Issue 3: 1585-1594. Doi: 10.3934/dcds.2018122

This issue Previous Article Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations Next Article Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian

Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus

1.
School of Mathematics and Statistics, Central South University, Changsha, China
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China
3.
Department of Mathematics, California State University Northridge, Northridge, CA 91330, United States

* Corresponding author: yaorf5812@stu.xjtu.edu.cn
* Corresponding author: yaorf5812@stu.xjtu.edu.cn

Received: June 30, 2017

Revised: October 31, 2017

Published: December 2018

The first author is supported by Tian Yuan Special Funds of the National Science Foundation of China (No.11626182)

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Abstract

In this paper, we show the following equation

$\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ u = 0&\text{ on }\partialΩ, \end{cases}$

has at most one positive radial solution for a certain range of $λ>0$. Here $p>1$ and $Ω$ is the annulus $\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$, $0<a<b$. We also show this solution is radially non-degenerate via the bifurcation methods.

Keywords:

Mathematics Subject Classification: 35J61, 35J25, 35B32.

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