Three solutions for a mixed boundary value problem involving the one-dimensional \(p\)-Laplacian
type: Item
instance of: scholarly article
title: Three solutions for a mixed boundary value problem involving the one-dimensional \(p\)-Laplacian
zbMATH Open document ID: 1080.34007
author: Diego Averna
author: Roberta Salvati
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-11-23T00:00:00Z
review text: This paper deals with two mixed boundary value problems of the form \[ -(| u'| ^{p-2}u')'=\lambda f(t,u), \quad u(a)=0=u'(b), \] and \[ -(| u'| ^{p-2}u')'+| u| ^{p-2}u=\lambda f(t,u), \quad u(a)=0=u'(b), \] where \(f\) is \(L^1\)-Carathéodory, \(p>1\) and \(\lambda >0\). Sufficient conditions are given which guarantee the existence of at least three solutions. For example, the following corollary is proved: ``Let \(f\) be positive, bounded and such that \(8 \int_0^{1/2} f(\xi) d\xi < {1 \over 3} \int_0^1 f(\xi) d\xi\). Then the problem \(-u''= f(u), \quad u(0)=0=u'(1)\), has at least three positive solutions.'' The proof is performed by applying an abstract three critical point theorem due to the first author and \textit{G. Bonanno} [Topol. Methods Nonlinear Anal. 22, No. 1, 93--103 (2003; Zbl 1048.58005)].
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2117739
zbMATH Keywords: mixed boundary value problem
zbMATH Keywords: three critical points theorem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.05.012
OpenAlex ID: W2129344909
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DOI: 10.1016/J.JMAA.2004.05.012
description: scientific article; zbMATH DE number 2117739