An introduction to nonlinear functional analysis and elliptic problems
type: Item
instance of: scholarly article
title: An introduction to nonlinear functional analysis and elliptic problems
zbMATH Open document ID: 1228.46001
DOI: 10.1007/978-0-8176-8114-2
author: A. Ambrosetti
author: David Arcoya
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 2011-06-30T00:00:00Z
review text: This book provides some basic abstract tools used in modern nonlinear analysis in strong relationship with their applications to semilinear elliptic boundary value problems. The content of the book is divided into two parts, which contain 13 chapters. In the first part, key results are discussed such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray-Schauder degree, critical point theory, and bifurcation theory. The second part of this volume shows how these abstract results apply to Dirichlet elliptic boundary value problems. A more detailed description is provided in what follows. The first chapter deals with general results, such as Fréchet differentiability, Nemytskij operators, weak and strong solutions of the linear Laplace equation, linear compact operators and their eigenvalues, and Sobolev spaces. Chapter 2 is concerned with the Banach contraction principle and with a fixed point theorem for increasing operators. In Chapter 3, some basic results are discussed: the local inversion theorem, the Hadamard-Caccioppoli global inversion theorem and the case in which the map to be inverted has fold singularities. Chapter 4 deals with the Leray-Schauder topological degree. Variational methods are discussed in the next chapter of this volume. The authors are mainly concerned with minima, the mountain pass theorem and the linking theorem. Chapter 6 is concerned with local and global bifurcation theory. The abstract results collected in the first part of the book are applied in the second part to prove existence and multiplicity results for semilinear elliptic Dirichlet boundary value problems on bounded domains in \({\mathbb R}^N\). The authors point out that the choice of the appropriate abstract tool depends on the behavior of the nonlinearity as well as on the kind of results one expects. In Chapter 7, it is argued how a semilinear elliptic boundary value problem can be transformed into an …
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 46-01
Mathematics Subject Classification ID: 47-01
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 35-01
zbMATH DE Number: 5915627
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: Leray-Schauder topological degree
zbMATH Keywords: bifurcation theory
zbMATH Keywords: critical points
zbMATH Keywords: fixed point theorem
zbMATH Keywords: global inversion theorems
zbMATH Keywords: nonlinear functional analysis
zbMATH Keywords: quasilinear problems
zbMATH Keywords: superlinear problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-8114-2
OpenAlex ID: W422968047
description: scientific article; zbMATH DE number 5915627