Some surprising results on a one-dimensional elliptic boundary value blow-up problem
type: Item
instance of: scholarly article
title: Some surprising results on a one-dimensional elliptic boundary value blow-up problem
zbMATH Open document ID: 0945.34007
DOI: 10.4171/ZAA/896
author: Yanyan Li
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2000-09-19T00:00:00Z
full work available at URL: https://eudml.org/doc/48521
review text: The paper is concerned with the existence of sign-changing solutions of the boundary blow-up problem \[ \Delta_{p}u=\lambda f(u), \quad u(a)=u(b)=+\infty, \tag{*} \] where \(\lambda\) is a positive parameter, \(f\geq 0\) and \(\Delta_{p}u\) is the 1-dimensional \(p\)-Laplacian on \((a,b)\) (i.e., \(\Delta_{p}u = (|u'(t)|^{p-2}u'(t))'\), \(a<t<b\)). The author studies, for some classes of functions \(f\), how the number of sign-changing solutions varies with \(\lambda\). A sample result: Let \(p=2\) (i.e., \(\Delta_{p}u=u''\)), \(f(u)=\alpha u^q\) for \(u\geq 0\) and \(f(u) = (1+\beta\sin|u|)|u|\) for \(u\leq 0\), with \(q>3\) and \(0<\beta<1\). Then there exist \(\lambda_{1}\), \(\lambda_{2}\) such that (*) has at least one sign-changing solution if \(\lambda\geq\lambda_{1}\), exactly one sign-changing solution if \(\lambda>\lambda_{2}\) and no sign-changing solutions if \(\lambda<\lambda_{1}\). Furthermore, for each \(n\geq 1\) there is \(\delta>0\) such that (*) has at least \(n\) sign-changing solutions for \(\lambda\in (\pi^2/(b-a)^2-\delta, \pi^2/(b-a)^2+\delta)\). In particular, there are infinitely many such solutions if \(\lambda = \pi^2/(b-a)^2\).
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1356393
zbMATH Keywords: boundary blow-up
zbMATH Keywords: multiplicity
zbMATH Keywords: concave and convex nonlinearity
MaRDI profile type: Publication
cites work: Existence of two boundary blow-up solutions for semilinear elliptic equations
cites work: Q4312129
cites work: Q4052566
cites work: Q4377130
cites work: Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up
OpenAlex ID: W2010905367
description: scientific article; zbMATH DE number 1356393