A uniqueness result for a Neumann problem involving the critical Sobolev exponent
type: Item
instance of: scholarly article
title: A uniqueness result for a Neumann problem involving the critical Sobolev exponent
zbMATH Open document ID: 1108.35059
author: Massimo Grossi
author: Chang-Shou Lin
author: S. Prashanth
published in: Mathematische Annalen
publication date: 2003-07-01T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\). The authors mainly interested in positive solutions of the Neumann problem for the following nonlinear elliptic equation: \[ -d^2\Delta u+u=u^p \quad\text{in }\Omega,\qquad u>0 \quad\text{in }\Omega, \qquad \frac{\partial u}{\partial n}=0 \quad\text{on }\partial\Omega, \tag{1} \] where \(d\) is a positive constant, \(p= \frac{N+2}{N-2}\), \(\Delta\) is the Laplacian and \(\nu\) stands for the unit outer normal to \(\partial\Omega\). In the case of small \(d>0\), the authors show the uniqueness (up to rotation) of the one-bubbling solution which concentrates at a point of the boundary.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1944287
zbMATH Keywords: uniqueness result
zbMATH Keywords: critical exponent
zbMATH Keywords: positive solution
zbMATH Keywords: one-bubbling solution
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: s00208-002-0393-6
OpenAlex ID: W1968202179
DOI: 10.1007/S00208-002-0393-6
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description: scientific article; zbMATH DE number 1944287