Minimax theorems
type: Item
instance of: scholarly article
title: Minimax theorems
zbMATH Open document ID: 0856.49001
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 1996-08-29T00:00:00Z
review text: In this interesting book a variety of minimax type theorems and theorems on minimizing and Palais-Smale sequences are presented. Among them I mention the mountain pass theorem of Ambrosetti-Rabinowitz, the saddle-point theorem and linking theorem of Rabinovitz, Ekeland's variational principle, quantitative deformation lemma of M. Willem, fountain theorem of Bartsch, and so on. They permit to prove the existence and multiplicity of the critical values of some functionals defined on Banach spaces and to characterize the critical points by means of the so-called Palais-Smale sequences, as well as to construct such sequences, \dots Some generalization of classical Lyusternik-Shnirelman category theory (using relative category defined by Reeken) is also given. The developed theory is applied with success to nonlinear PDE's in order to obtain the existence results; for instance, for semilinear Dirichlet problems (also with concave and convex nonlinearities), for semilinear Schrödinger equation (in this Section the Kryszewski-Szulkin degree theory is used), for generalized Kadomtsev-Petviashvili equation, where the existence of solitary waves is proved. The material of the book covers many recent tools and some unpublished results in the domain of nonlinear PDE's and their applications.
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 49K35
zbMATH DE Number: 921895
zbMATH Keywords: minimax theorems
zbMATH Keywords: critical values
zbMATH Keywords: nonlinear partial differential equations
zbMATH Keywords: Palais-Smale sequences
zbMATH Keywords: critical points
zbMATH Keywords: Lyusternik-Shnirelman category theory
zbMATH Keywords: degree theory
zbMATH Keywords: solitary waves
author: Michel Willem
reviewed by: Zdzisław Denkowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 921895