A mountain pass theorem for a suitable class of functions
type: Item
instance of: scholarly article
title: A mountain pass theorem for a suitable class of functions
zbMATH Open document ID: 1172.47060
DOI: 10.1216/RMJ-2009-39-3-707
author: Diego Averna
author: Gabriele Bonanno
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: In this article, a mountain pass theorem is obtained for a functional of the form \(J=\Phi-\Psi\) defined on a reflexive real Banach space, where \(\Phi\) and \(\Psi\) are continuously Gâteau differentiable functions, \(\Phi\) is convex, and \(J_M = \Phi-\Psi_M\) satisfies the Palais--Smale condition (PS)\(_c\) with \(\Psi_M\) the cut-off function at a suitable constant \(M\). This mountain pass theorem is combined to a result on the existence and localization of two local minima of \(J\) [\textit{G.\,Bonanno}, J.~Math.\ Anal.\ Appl.\ 299, No.\,2, 600--614 (2004; Zbl 1071.34015)] to deduce a three critical points result without assuming the coercivity of \(J\). This result is then applied to obtain the existence of three nonnegative solutions of two-point boundary value problems with no asymptotic conditions.
reviewed by: Marlène Frigon
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5566197
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: critical points
zbMATH Keywords: three solutions
zbMATH Keywords: two-point boundary value problem
MaRDI profile type: Publication
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full work available at URL: rmj-2009-39-3-707
OpenAlex ID: W2038921421
description: scientific article; zbMATH DE number 5566197