Positive solutions of the Robin problem for semilinear elliptic equations on annuli
type: Item
instance of: scholarly article
title: Positive solutions of the Robin problem for semilinear elliptic equations on annuli
zbMATH Open document ID: 1198.35096
DOI: 10.4171/RLM/516
author: Yu-Xia Fu
author: Qiuyi Dai
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2010-09-19T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Summary: Let \(n\geq 3\) and \(\Omega_R= \{x\in\mathbb R^n\); \(R<|x|<1\}\). We consider the following Robin problem: \[ \begin{cases} -\Delta u=f(u), &x\in\Omega_R,\\ u>0, &x\in\Omega_R,\\ \dfrac{\partial u}{\partial \nu}+\beta u=0 &x\in\partial\Omega_R, \end{cases} \] where \(\beta\) is a positive parameter and \(\nu\) is the unit outward vector normal to \(\partial\Omega_R\). Under some assumptions, we prove that the above problem has at most one solution when \(\beta\) is small enough, and under some other assumptions, the above problem has at least \(k\) nonradial solutions when \(\beta\) is large enough.
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 5787949
zbMATH Keywords: positive solutions
zbMATH Keywords: Robin problem
zbMATH Keywords: semilinear elliptic equations
MaRDI profile type: Publication
cites work: Dual variational methods in critical point theory and applications
cites work: On the method of moving planes and the sliding method
cites work: Q3745374
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: A Faber-Krahn inequality for Robin problems in any space dimension
cites work: A priori bounds for positive solutions of nonlinear elliptic equations
cites work: Monotonicity results for the principal eigenvalue of the generalized Robin problem
cites work: A prior estimate and existence of positive solutions of semilinear elliptic equation with the third boundary value problem
cites work: On the Schrödinger equation and the eigenvalue problem
cites work: Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus
cites work: Existence of Positive Nonradial Solutions for Nonlinear Elliptic Equations in Annular Domains
cites work: Uniqueness of solutions of nonlinear Dirichlet problems
OpenAlex ID: W2021825230
description: scientific article; zbMATH DE number 5787949