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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

MSC:
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
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[1] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349 – 381. · Zbl 0273.49063
[2] Herbert Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982), no. 4, 591 – 595. · Zbl 0501.58012
[3] Paul H. Rabinowitz, A note on topological degree for potential operators, J. Math. Anal. Appl. 51 (1975), no. 2, 483 – 492. · Zbl 0307.47058 · doi:10.1016/0022-247X(75)90134-1 · doi.org
[4] Helmut Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), no. 4, 493 – 514. · Zbl 0488.47034 · doi:10.1007/BF01457453 · doi.org
[5] Erich H. Rothe, A relation between the type numbers of a critical point and the index of the corresponding field of gradient vectors, Math. Nachr. 4 (1951), 12 – 17. · Zbl 0044.31903 · doi:10.1002/mana.19500040103 · doi.org
[6] Adele Manes and Anna Maria Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4) 7 (1973), 285 – 301 (Italian, with English summary). · Zbl 0275.49042
[7] Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999 – 1030. · Zbl 0477.35075 · doi:10.1080/03605308008820162 · doi.org
[8] P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Eigenvalues of non-linear problems (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1974) Edizioni Cremonese, Rome, 1974, pp. 139 – 195.
[9] Detlef Gromoll and Wolfgang Meyer, On differentiable functions with isolated critical points, Topology 8 (1969), 361 – 369. · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6 · doi.org
[10] Floris Takens, A note on sufficiency of jets, Invent. Math. 13 (1971), 225 – 231. · Zbl 0231.58008 · doi:10.1007/BF01404632 · doi.org
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