Uniqueness of positive solutions for a boundary blow-up problem
type: Item
instance of: scholarly article
title: Uniqueness of positive solutions for a boundary blow-up problem
zbMATH Open document ID: 1182.35006
author: Jorge García-Melian
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-06T00:00:00Z
review text: In this paper the author deals with the positive solutions to the boundary blow-up problem \[ \begin{cases} \Delta u=f(u) &\text{in }\Omega,\\ u=\infty &\text{on }\partial \Omega, \end{cases} \tag{1} \] where \(\Omega \) is a smooth bounded domain of \(\mathbb{R}^N\), \(f\) is a continuous function and a solution means a function \(u\in H_{\text{loc}}^1(\Omega )\cap C(\Omega )\) verifying the equation in \( (1)\) in the weak sense, and the boundary condition in the sense \(u(x)\rightarrow \infty \) as \(\text{dist}(x,\partial \Omega )\rightarrow 0\). First, the author shows that if \(f\) is increasing and there exists \(p>1\) such that \(\frac{f(t)}{t^p}\) is increasing for large \(t\), then \( (1)\) admits a unique positive solution. Then, they extend this uniqueness result to slightly more general problem than \((1)\), involving weights, namely \[ \begin{cases} \Delta u=a(x) f(u) &\text{in }\Omega,\\ u=\infty &\text{on }\partial\Omega, \end{cases} \] where \(a\in C(\overline{\Omega })\) is a nonnegative function, possibly vanishing on \(\partial \Omega\).
reviewed by: Cristian Chifu
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5626929
zbMATH Keywords: boundary blow-up
zbMATH Keywords: uniqueness
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: semilinear elliptic equation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.06.077
OpenAlex ID: W2025242019
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DOI: 10.1016/J.JMAA.2009.06.077
description: scientific article; zbMATH DE number 5626929