Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems.

*(English)*Zbl 0746.49010
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.

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 |