Uniqueness results through a priori estimates. II: Dirichlet problem
type: Item
instance of: scholarly article
title: Uniqueness results through a priori estimates. II: Dirichlet problem
zbMATH Open document ID: 0965.35062
author: Meijun Zhu
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-07-22T00:00:00Z
review text: The author establishes a priori bounds and uniqueness for positive solutions of \[ -\Delta u-au= u^{{n+2}\over{n-2}}\quad\text{in }\Omega, \qquad u|_{\partial\Omega}\equiv 0,\tag{1} \] for bounded domains \(\Omega\) in \({\mathbb R}^n\) (\(n\geq 3\)) with smooth boundary and \(0<a<\lambda_1\) (\(\lambda_1\) the principal eigenvalue of \(-\Delta\) on \(\Omega\) subject to \(u|_{\partial\Omega}\equiv 0\)). He shows: if \(u_a\) is a positive solution of (1) for \(0<a<\lambda_1\), then \(\|u_a\|_\infty\to 0\) as \(a\to\lambda_1\) and there exists a \(\underline a\in(0,\lambda_1)\) such that \(u_a\) is the only positive solution of (1) for \(\underline a<a<\lambda_1\). The proof relies on reasoning by contradiction, and buzzwords such as energy independent bounds and blow-up points may be an incentive to read the paper which extends classical work of \textit{F. V. Atkinson} and \textit{L. A. Peletier} [Nonlinear Anal., Theory Methods Appl. 10, 755-776 (1986; Zbl 0662.34024)] in the case of \(\Omega\) being the unit ball.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1506525
zbMATH Keywords: positive solutions
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: first eigenvalue
zbMATH Keywords: uniqueness
zbMATH Keywords: uniform \(L^\infty\) a priori estimates
reviewed by: Georg Hetzer
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6884
OpenAlex ID: W2013208753
cites work: Emden-Fowler equations involving critical exponents
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Symmetry and related properties via the maximum principle
cites work: Global and local behavior of positive solutions of nonlinear elliptic equations
cites work: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent
cites work: Remarks on some quasilinear elliptic equations
cites work: Prescribing scalar curvature on \(\mathbb{S}^ n\) and related problems. I
cites work: Prescribing scalar curvature on Sn and related problems, part II: Existence and compactness
cites work: YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS
cites work: Q3359340
cites work: Prescribed scalar curvature on the \(n\)-sphere
cites work: Uniqueness results through a priori estimates. I: A three dimensional Neumann problem
DOI: 10.1006/JMAA.2000.6884
description: scientific article; zbMATH DE number 1506525