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The homotopy index and partial differential equations. (English) Zbl 0628.58006
Universitext. Berlin etc.: Springer-Verlag. XII, 208 p.; DM 69.00 (1987).
This book is a well written introduction to the Conley homotopy index and its applications to problems involving partial differential equations. In the first chapter the author introduces the homotopy index and discusses its properties. Two further chapters are devoted to applications. In particular the author deals with asymptotically linear systems, he gives estimates at zero and nontrivial solutions of elliptic equations, positive heteroclinic orbits of second-order parabolic equations are treated, and finally in this chapter, a homotopy index continuation method and periodic solutions of second-order gradient systems are considered. Chapter III, entitled “Selected Topics”, gives further applications, for example the homotopy index is discussed in its relation to the Morse theory on Hilbert manifolds.
The book ends with bibliographical notes and comments.
Reviewer: N.Jacob

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B40 Asymptotic behavior of solutions to PDEs
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations