Ambrosetti A., Arcoya Álvarez D. An Introduction to Nonlinear Functional Analysis and Elliptic Problems

Birkhäuser Basel, 2011. — 199 p. — ISBN: 978-0-8176-8114-2 (eBook), 978-0-8176-8113-5 (Hardcover).This self-contained textbook provides the basic, abstracttoolsused innonlinear analysisand their applications to semilinear elliptic boundary value problems.By firstoutlining the advantages and disadvantages of each method, this comprehensive textdisplays how variousapproachescan easily beappliedto a range of model cases.
An Introduction to Nonlinear Functional Analysis and Elliptic Problemsis divided into two parts: the first discusses keyresults 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 shows how these abstract results apply to Dirichlet elliptic boundary value problems. The exposition is driven by numerous prototype problems and exposes a variety of approaches tosolving them.
Complete with a preliminary chapter, an appendix that includes further results on weak derivatives, and chapter-by-chapter exercises, this book is apractical text for an introductory course or seminar on nonlinear functional analysis.Preliminaries.
Some Fixed Point Theorems.
Local and Global Inversion Theorems.
Leray—Schauder Topological Degree.
An Outline of Critical Points.
Bifurcation Theory.
Elliptic Problems and Functional Analysis.
Problems with A Priori Bounds.
Asymptotically Linear Problems.
Asymmetric Nonlinearities.
Superlinear Problems.
Quasilinear Problems.
Stationary States of Evolution Equations.