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A note on the topological degree at a critical point of mountainpass- type. (English) Zbl 0545.58015
In this interesting paper the author proves that, under a technical condition satisfied in many applications, the Leray-Schauder index of a critical point of mountainpass type is -1. (Of course the gradient of the corresponding functional has the form identity-compact.) This result is important in the study of semi-linear second order elliptic problems [the author, Math. Ann. 261, 493-514 (1982; Zbl 0488.47034)]. Moreover it is easy to compute the critical groups of the mountainpass point from the proof, so that Morse theory is applicable [K. C. Chang, Applications of homology theory to some problems in differential equations, preprint (1983)]. The proof depends on a generalization to \(C^ 2\) functions of the Gromoll-Meyer degenerate Morse lemma. The homological characterization of the mountainpass point has been obtained independently by G. Tian [Chin. Bull. Sci., to appear].
Reviewer: M.Willem

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J05 Equations involving nonlinear operators (general)
58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
34G20 Nonlinear differential equations in abstract spaces
Full Text: DOI
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