On the uniqueness of solutions of a semilinear equation in an annulus

Carmen Cortázar; Marta García-Huidobro; Pilar Herreros; Satoshi Tanaka

doi:10.3934/cpaa.2021029

Article Contents

2021, Volume 20, Issue 4: 1479-1496. Doi: 10.3934/cpaa.2021029

This issue Previous Article Asymptotics for the higher-order derivative nonlinear Schrödinger equation Next Article An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well

On the uniqueness of solutions of a semilinear equation in an annulus

1.
Departamento de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
2.
Mathematical Institute, Tohoku University, Aoba 6–3, Aramaki, Aoba-ku, Sendai 980–8578, Japan

^* Corresponding author
^* Corresponding author

Received: September 2020

Revised: January 2021

Early access: March 2021

Published: April 2021

This research was supported by FONDECYT-1190102 for the first and second author, and FONDECYT- 1170665 for the third author and by JSPS KAKENHI Grant Number 19K03595 and 17H01095 for the fourth author

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Abstract

We establish the uniqueness of positive radial solutions of

$ \begin{align} \begin{cases} \Delta u +f(u) = 0, \quad x\in A \\ u(x) = 0 \qquad \qquad x\in \partial A \end{cases} \;\;\;\; (P)\end{align} $

where $ A: = A_{a, b} = \{ x\in {\mathbb R}^n : a<|x|<b \} $, $ 0<a<b\le\infty $. We assume that the nonlinearity $ f\in C[0, \infty)\cap C^1(0, \infty) $ is such that $ f(0) = 0 $ and satisfies some convexity and growth conditions, and either $ f(s)>0 $ for all $ s>0 $, or has one zero at $ B>0 $, is non positive and not identically 0 in $ (0, B) $ and it is positive in $ (B, \infty) $.

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Mathematics Subject Classification: Primary: 35J61, 35A02; Secondary: 35A24.

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References

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