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.

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)

##### MSC:

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 |