Infinite dimensional Morse theory and multiple solution problems
type: Item
instance of: scholarly article
title: Infinite dimensional Morse theory and multiple solution problems
zbMATH Open document ID: 0779.58005
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 1993-06-27T00:00:00Z
review text: This excellent monograph deals with a new direction in modern mathematics: critical point theory as a way of studying multiple solutions of differential equations which arise in the calculus of variations. It represents a more completed version of the author's book ``Infinite dimensional Morse theory and its applications'' (1985; Zbl 0609.58001). The book contains five chapters and, taking into account the diversity of the exposed material, I prefer to present separately each of them. Chapter I is ``Infinite Dimensional Morse Theory'' and it presents basic notions and results concerning: pseudo gradient vector field and the deformation theorems, critical groups and Morse type numbers, Gromoll- Meyer theory, extension of Morse theory (Morse theory under general boundary conditions, Morse theory on locally convex closed set), equivariant Morse theory. This chapter also contains some preliminaries on algebraic topology and on infinite dimensional manifolds, which are very useful in the understanding of the included material. Chapter II is entitled ``Critical Point Theory'' and it contains six sections related to the following aspects: topological link, Morse indices and minimax critical points, connections with other theories (Leray-Schauder degree theory, Lyusternik-Shnirelman theory, relative category), invariant functionals, some abstract critical point theorems, perturbation theory (perturbation on critical manifolds, Uhlenbeck's perturbation method). A special attention is paid to some theorems obtained by \textit{A. Ambrosetti} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)], which are proved in this chapter via homology methods. In Chapter III, ``Applications to Semilinear Elliptic Boundary Value Problems'', one presents applications of critical point theory. Some very interesting results concerning applications to problems with superlinear, asymptotically linear and bounded nonlinear terms are given. Chapter IV is ``Multiple P…
reviewed by: Dorin Andrica
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 58Exx
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 58-01
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 217439
zbMATH Keywords: Morse theory
zbMATH Keywords: critical point theory
zbMATH Keywords: multiple solutions of differential equations
zbMATH Keywords: variations
author: Kung Ching Chang
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 217439