Semilinear elliptic equations for beginners. Existence results via the variational approach. (English) Zbl 1214.35025

Badiale, Marino; Serra, Enrico

Universitext. London: Springer (ISBN 978-0-85729-226-1/pbk; 978-0-85729-227-8/ebook). x, 199 p. (2011).

The aim of this book is to introduce the reader to different topics of the theory of semilinear elliptic partial differential equations by avoiding technicalities and refinements. The authors give a systematic treatment of the basic mathematical theory and constructive methods for nonlinear elliptic differential equations, as well as their applications to various problems. The variational approach to these equations has experienced spectacular success in recent years, reaching a high level of complexity and refinement, with various applications to mathematical physics, geometry, economics, and other fields. Additionally, some of the simplest variational methods are evolving as classical tools in the field of nonlinear differential equations.
The content of this volume is divided into four chapters, as follows: 1. Introduction and basic results; 2. Minimization techniques: compact problems; 3. Minimization techniques: Lack of compactness; 4. Introduction to minimax methods.
The first chapter contains a review of differential calculus for nonlinear operators, with many examples, and a few basic facts from the linear theory that are used in the book. Chapter 2 introduces the fundamentals of minimization techniques, which are very useful to obtain critical points of the associated energy functionals. Typical examples of such sets are spheres or the Nehari manifold. The unifying feature of this chapter is the fact that all problems share some compactness properties that simplify the convergence arguments. Chapter 3 is still devoted to minimization techniques, but this time the examples are all taken from problems that lack compactness. Chapter 4 introduces the reader to some more refined variational methods, where minimization is replaced by minimax procedures. The authors are concerned with the mountain pass theorem and the saddle point theorem, each discussed with applications to the specific problems that motivate them.
This book is a valuable reference book for specialists in the field and an excellent graduate text giving an overview of the literature on solutions of semilinear elliptic equations. In the reviewer’s opinion, the book should be strongly recommended to anyone, either graduate student or researcher, who is interested in variational methods and their applications to partial differential equations of elliptic type.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

MSC:

35J61	Semilinear elliptic equations
35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35J87	Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
35J20	Variational methods for second-order elliptic equations
35D30	Weak solutions to PDEs
49J40	Variational inequalities
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

semilinear elliptic equation; critical point theory; variational method; weak solution