Introduction à la théorie des points critiques et applications aux problèmes elliptiques. (French) Zbl 0797.58005

Mathématiques & Applications (Berlin). 13. Paris: Springer-Verlag,. viii, 325p. (1993).
This book is intended as a higher level course in nonlinear analysis and its applications to differential equations.
In the first chapter, the author revises a number of results on linear analysis and partial differential equations. He also extends some finite- dimensional results to infinite dimensions. In Chapter 2, he introduces the Brouwer degree in finite dimensions and the Leray-Schauder degree in infinite dimensions. He also gives some applications to nonlinear elliptic partial differential equations. In Chapter 3, he discusses critical point theory and applications. He also discusses Ky-Fan type theorems. In Chapter 4, he discusses constrained variational problems, including Ljusternik-Schnirelman theory. In Chapter 5, he discusses the variational problems which are not symmetric. He includes a discussion of perturbations of odd mappings and jumping nonlinearities. Lastly, in Chapter 6, he discusses variational problems where the Palais-Smale condition fails.
The book has many interesting applications and many interesting exercises. I would recommend it for a course though the students will need to be well prepared.
Reviewer: E.Dancer (Sydney)


58-02 Research exposition (monographs, survey articles) pertaining to global analysis
47J05 Equations involving nonlinear operators (general)
35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations