Classification of balanced sets and critical points of even functions on spheres
HTML articles powered by AMS MathViewer
- by Charles V. Coffman PDF
- Trans. Amer. Math. Soc. 326 (1991), 727-747 Request permission
Abstract:
The Lyusternik-Schnirelman approach to the study of critical points of even functionals on the sphere ${S^N}$ employs min-max or max-min principles whose formulation uses a numerical invariant that is defined for compact balanced subsets of ${S^N}$. The Krasnosel’skii genus is an example. Here we study a general class of such invariants (which is quite large) with particular attention to the following questions: formulation of dual variational principles, multiplicity results for critical points, and determination of the Morse index of nondegenerate critical points.References
- Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7 A. Bahri, Une méthode perturbative en théorie de Morse, Thése d’Etat, Publications de l’Université Paris VI, 1981.
- Abbas Bahri and Henri Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1–32. MR 621969, DOI 10.1090/S0002-9947-1981-0621969-9
- Abbas Bahri and Pierre-Louis Lions, Remarques sur la théorie variationnelle des points critiques et applications, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 5, 145–147 (French, with English summary). MR 801948
- A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), no. 8, 1027–1037. MR 968487, DOI 10.1002/cpa.3160410803
- Charles V. Coffman, Lyusternik-Schnirelman theory: complementary principles and the Morse index, Nonlinear Anal. 12 (1988), no. 5, 507–529. MR 940607, DOI 10.1016/0362-546X(88)90046-6
- P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416–441. MR 163310, DOI 10.1090/S0002-9904-1960-10492-2
- Edward R. Fadell and Paul H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Functional Analysis 26 (1977), no. 1, 48–67. MR 0448409, DOI 10.1016/0022-1236(77)90015-5
- Hans-Peter Heinz, Un principe de maximum-minimum pour les valeurs critiques d’une fonctionnelle non linéaire, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1317–A1318 (French). MR 309149
- M. A. Krasnosel′skiĭ, On the estimation of the number of critical points of functionals, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 2(48), 157–164 (Russian). MR 0048699
- M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. MR 0159197
- A. Marino and G. Prodi, Metodi perturbativi nella teoria di Morse, Boll. Un. Mat. Ital. (4) 11 (1975), no. 3, suppl., 1–32 (Italian, with English summary). Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday. MR 0418150
- Paul H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202. MR 320850, DOI 10.1216/RMJ-1973-3-2-161
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Chung-Tao Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I, Ann. of Math. (2) 60 (1954), 262–282. MR 65910, DOI 10.2307/1969632
- Chung-Tao Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. II, Ann. of Math. (2) 62 (1955), 271–283. MR 72470, DOI 10.2307/1969681
- Chung-Tao Yang, Continuous functions from spheres to euclidean spaces, Ann of Math. (2) 62 (1955), 284–292. MR 0072471, DOI 10.2307/1969682
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 326 (1991), 727-747
- MSC: Primary 58E05; Secondary 47H99, 55M99
- DOI: https://doi.org/10.1090/S0002-9947-1991-1007802-0
- MathSciNet review: 1007802