A remark on the existence of explosive solutions for a class of semilinear elliptic equations
type: Item
instance of: scholarly article
title: A remark on the existence of explosive solutions for a class of semilinear elliptic equations
zbMATH Open document ID: 0964.35053
DOI: 10.1016/S0362-546X(98)00270-3
author: Zhijun Zhang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-07-02T00:00:00Z
review text: There has been considerable research on the problem of finding a positive solution \(u\) of the differential equation \(\Delta u = k(x)f(u)\) in \(\Omega\), a bounded domain in \(\mathbb R^n\), which diverges to infinity at \(\partial\Omega\) when the function \(k\) is a positive constant and \(f(u) = e^u\) or \(f(u) = u^p\) for some constant \(p>1\). In this work, the author shows that this problem has a solution for a class of nonconstant functions \(k\). The main conditions are that \(k\) is uniformly bounded from below in \(\Omega\) and does not blow up too fast near \(\partial \Omega\). Specifically either \(k(x) = O(d(x)^{\beta-2})\) for some \(\beta \in (0,2]\), where \(d\) is distance to the boundary, or \(d \in L^q\) for some \(q > n/2\). The key step is to approximate the domain \(\Omega\) from within by a sequence of smooth domains (so that \(k\) is uniformly bounded on the approximating domains). The maximum principle shows that the sequence of solutions of the approximating problems with infinite boundary data (which are known to exist) is decreasing and a simple change of dependent variable along with the maximum principle shows that this sequence is bounded from below by a strictly positive function.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1472250
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: explosive solutions
zbMATH Keywords: existence
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00270-3
OpenAlex ID: W1970946442
description: scientific article; zbMATH DE number 1472250