Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. (English) Zbl 0746.49010

Struwe, Michael

Berlin etc.: Springer-Verlag. xiv, 244 p. (1990).

This is an interesting and readable book on modern results by variational methods. It links variational ideas with global analysis and the work of Lyusternik and Schnirelman, Morse, Conley, Zehnder, to mention a few names. We give an idea of its contents.
In Chapter 1 direct methods are reviewed with some recent extensions like Ekeland’s variational principle. Chapter 2 deals with minimax methods involving index theory and critical points; this chapter extensively applies Lyusternik-Schnirelman theory with applications to Hamiltonian systems and semilinear elliptic equations. Chapter 3 again focuses on minimax methods with the emphasis on limit cases of the Palais-Smale conditions. In three appendices a number of relevant concepts of modern analysis are discussed.
Although the author does not claim completeness of his references of the modern literature, the text provides a thorough introduction to research as it is going on at the moment on variational methods for differential equations. By its nature the book is concerned with qualitative theory which requires abstract theory, however it has been written in a fluent style which makes it accessible to a wide audience of mathematicians starting with the level of a good graduate student.

Reviewer: F. Verhulst (Utrecht)

MSC:

49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
34C25	Periodic solutions to ordinary differential equations
35A15	Variational methods applied to PDEs
35F20	Nonlinear first-order PDEs
37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
47J30	Variational methods involving nonlinear operators
49J10	Existence theories for free problems in two or more independent variables
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

Ekeland’s variational principle; minimax methods; index theory; critical points; Lyusternik-Schnirelman theory; Hamiltonian systems; semilinear elliptic equations