Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations
type: Item
instance of: scholarly article
title: Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations
zbMATH Open document ID: 0877.35042
DOI: 10.1016/S0294-1449(97)80146-1
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1997-05-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78410
review text: The authors study properties of positive solutions of \[ \Delta u+ hu-ku^p= f\tag{1} \] in a (possibly) nonsmooth \(N\)-dimensional domain \(\Omega\), \(N\geq 2\), subject to the condition \[ u(x)\to\infty\quad\text{if}\quad \delta(x):= \text{dist}(x,\partial\Omega)\to 0.\tag{2} \] Here \(p>1\) and \(h\), \(k\), \(f\) are continuous in \(\overline\Omega\) with \(k>0\) and \(f\geq 0\). Positive solutions of (1) satisfying (2) are called large solutions. A central point of this paper is the following localization principle: let \(\Omega\) be a (not necessarily bounded) domain having the graph property and suppose \(u\) is a positive solution of (1) satisfying \(u(x)\to\infty\) locally uniformly as \(x\to\Gamma\), where \(\Gamma\subset\partial\Omega\) is relatively open. If \(v\) is a large solution, then \(v(x)/u(x)\to 1\) locally uniformly as \(x\to\Gamma\). Closely related to this is a uniqueness result for large solutions in bounded domains having the graph property. For bounded Lipschitz domains the authors prove the existence of positive constants \(c_1\leq c_2\) such that the (unique) large solution \(u\) of (1) satisfies \(c_1\delta(x)^{-{2\over p-1}}\leq u(x)\leq c_2\delta(x)^{-{2\over p-1}}\) for all \(x\in\Omega\). This is also a consequence of the localization principle and an existence theorem, obtained for large solutions in bounded domains satisfying the exterior cone condition. If the domain is not Lipschitz, the rate of blow-up at the boundary may be lower. This is proved for domains having a re-entrant cusp in the case \(p\geq(N-1)/(N-3)\). Finally, the authors discuss the dependence of large solutions on the function \(k\) and the domain \(\Omega\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J67
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1013373
zbMATH Keywords: uniqueness for large solutions
zbMATH Keywords: large solutions
zbMATH Keywords: localization principle
zbMATH Keywords: rate of blow-up at the boundary
author: Moshe Marcus
author: Laurent Véron
reviewed by: Ralf Beyerstedt
MaRDI profile type: Publication
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OpenAlex ID: W2021894352
description: scientific article; zbMATH DE number 1013373