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Morse theory in nonlinear analysis. (English) Zbl 0960.58006
Ambrosetti, A. (ed.) et al., Proceedings of the 2nd school on nonlinear functional analysis and applications to differential equations, ICTP, Trieste, Italy, April 21-May 9, 1997. Singapore: World Scientific. 60-101 (1998).
The paper is a survey of infinite dimensional Morse theory for smooth functionals on Finsler manifolds and its applications to nonlinear elliptic equations. The author presents basic concepts of Morse theory, like deformation techniques, the notions of Morse critical groups and Gromoll-Meyer pair, Morse inequalities etc., as well as some recent generalizations. The last part of the paper is devoted to applications. In particular, the author describes results of T. Bartsch and Z.-Q. Wang [Topol. Methods Nonlinear Anal. 7, No. 1, 115-131 (1996; Zbl 0903.58004)] on the existence of changing sign solutions of semilinear elliptic problems.
The survey is written in a nice pedagogical way, with careful proofs of main results. It can be highly recommended for beginners in the field as a parallel reading to the author’s book “Infinite dimensional Morse theory and multiple solution problems” (1993; Zbl 0779.58005).
For the entire collection see [Zbl 0941.00024].

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
47H11 Degree theory for nonlinear operators
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35J60 Nonlinear elliptic equations