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Infinite dimensional Morse theory and multiple solution problems. (English) Zbl 0779.58005
Progress in Nonlinear Differential Equations and their Applications. 6. Boston: Birkhäuser. x, 312 p. (1993).
This excellent monograph deals with a new direction in modern mathematics: critical point theory as a way of studying multiple solutions of differential equations which arise in the calculus of variations. It represents a more completed version of the author’s book “Infinite dimensional Morse theory and its applications” (1985; Zbl 0609.58001).
The book contains five chapters and, taking into account the diversity of the exposed material, I prefer to present separately each of them. Chapter I is “Infinite Dimensional Morse Theory” and it presents basic notions and results concerning: pseudo gradient vector field and the deformation theorems, critical groups and Morse type numbers, Gromoll- Meyer theory, extension of Morse theory (Morse theory under general boundary conditions, Morse theory on locally convex closed set), equivariant Morse theory. This chapter also contains some preliminaries on algebraic topology and on infinite dimensional manifolds, which are very useful in the understanding of the included material. Chapter II is entitled “Critical Point Theory” and it contains six sections related to the following aspects: topological link, Morse indices and minimax critical points, connections with other theories (Leray-Schauder degree theory, Lyusternik-Shnirelman theory, relative category), invariant functionals, some abstract critical point theorems, perturbation theory (perturbation on critical manifolds, Uhlenbeck’s perturbation method). A special attention is paid to some theorems obtained by A. Ambrosetti and P. H. Rabinowitz [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)], which are proved in this chapter via homology methods. In Chapter III, “Applications to Semilinear Elliptic Boundary Value Problems”, one presents applications of critical point theory. Some very interesting results concerning applications to problems with superlinear, asymptotically linear and bounded nonlinear terms are given. Chapter IV is “Multiple Periodic Solutions of Hamiltonian Systems” and it deals with applications of Morse theory to estimate the number of solutions of Hamiltonian systems. The main problems studied in this chapter are the following: asymptotically linear systems, Hamiltonians with periodic nonlinearities, second order systems with singular potentials, the double pendulum equation, Arnold conjectures on symplectic fixed points and on Lagrangian intersections. In Chapter V, “Applications to Harmonic Maps and Minimal Surfaces”, very interesting applications of infinite dimensional Morse theory in the study of some geometric variational problems are included. The author has in view the following aspects: harmonic maps and heat flow, the existence and multiplicity for harmonic maps, the Plateau problem for minimal surfaces.
The book also contains an Appendix presenting in a self-contained way the famous proof of the Morse inequalities obtained by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)].
The book ends with a rich bibliography containing 232 titles and with a useful subject index. It is written in a very clear and rigorous manner and it is recommended for researchers and graduate students working in nonlinear analysis, nonlinear functional analysis, partial differential equations, ordinary differential equations, differential geometry and topology.

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58Exx Variational problems in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces