Minimax methods in critical point theory with applications to differential equations. (English) Zbl 0609.58002

Rabinowitz, Paul H.

Regional Conference Series in Mathematics 65. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0715-3). viii, 100 p. (1986).

The author provides an essentially self-contained introduction to minimax methods in abstract critical point theory. He also considers applications of minimax methods to problems in differential equations, especially to semilinear elliptic partial differential equations and periodic solutions of Hamiltonian systems. First of all, the cornerstones of the theory, viz. the Mountain pass theorem and the saddle point theorem and variants of them are treated in detail. All these theorems require the so-called Palais-Smale condition to be satisfied. Another part of the monograph is devoted to the study of variational problems in which symmetries play a role. Using index theories the author investigates the existence of multiple critical points of symmetric functionals. Here both constrained and unconstrained variational problems are treated. He also studies an example from partial differential equations in which a symmetric functional is subjected to a perturbation which destroys the symmetry. Finally, applications of minimax methods to bifurcation problems are considered. Two appendices are devoted to the more technical details. So the so-called deformation theorem is proved which is basic for the proof of the mountain pass theorem. Furthermore, some results are given which are useful in verifying abstract conditions like the Palais-Smale condition in a partial differential equations setting.

Reviewer: B.Schomburg

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E07	Variational problems in abstract bifurcation theory in infinite-dimensional spaces
35J60	Nonlinear elliptic equations
47J05	Equations involving nonlinear operators (general)
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

minimax methods; abstract critical point theory; semilinear elliptic partial differential equations; periodic solutions of Hamiltonian systems; Mountain pass theorem; saddle point theorem; Palais-Smale condition; symmetries; index theories; existence of multiple critical points of symmetric functionals; bifurcation problems; deformation theorem