A min-max principle for non-differentiable functions with a weak compactness condition
type: Item
instance of: scholarly article
title: A min-max principle for non-differentiable functions with a weak compactness condition
zbMATH Open document ID: 1208.58014
author: R. Livrea
author: Salvatore A. Marano
published in: Communications on Pure and Applied Analysis
publication date: 2009-04-23T00:00:00Z
review text: The authors extend the min-max principle of \textit{N. Ghoussoub} [Proc. Am. Math. Soc. 117, No. 2, 439--447 (1993; Zbl 0791.49028)] to local Lipschitz continuous functions satisfying a weak (PS) condition, which induces both the non-smooth (PS)\(^+\) or (PS)\(^-\) condition [Definition 2 of \textit{K.-C. Chang}, J. Math. Anal. Appl. 80, 102--129 (1981; Zbl 0487.49027)] and non-smooth Cerami condition. The main result i.e., Theorem 3.1 is proved by adapting Ghoussoub's approach to the non-smooth framework and exploiting Ekeland's variational principle with a suitable metric of geodesic type, and argument of non-smooth analysis. The authors also give two direct consequences of Theorem 3.1, i.e., Theorems 3.2 and 3.3. Theorem 3.2 provides a more refined version of several recent critical point results and Theorem 3.3 might reveal useful to get non-smooth variants of structure or multiplicity results for \(C^1\) functions.
reviewed by: Duanzhi Zhang
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 5546292
zbMATH Keywords: critical points
zbMATH Keywords: locally Lipshitz continuous functions
zbMATH Keywords: weak Palais-Smale condition
zbMATH Keywords: mountain pass geometry
MaRDI profile type: Publication
full work available at URL: cpaa.2009.8.1019
OpenAlex ID: W1973570915
DOI: 10.3934/CPAA.2009.8.1019
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description: scientific article; zbMATH DE number 5546292