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On critical point theory for indefinite functionals in the presence of symmetries


Author: Vieri Benci
Journal: Trans. Amer. Math. Soc. 274 (1982), 533-572
MSC: Primary 58E05; Secondary 58F22, 70H05
MathSciNet review: 675067
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Abstract: We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group.

We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments.

The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited.


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  • [1] H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 4, 539–603. MR 600524
  • [2] Herbert Amann and Eduard Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math. 32 (1980), no. 1-2, 149–189. MR 592715, 10.1007/BF01298187
  • [3] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183
  • [4] P. Bartolo, V. Benci, and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981–1012. MR 713209, 10.1016/0362-546X(83)90115-3
  • [5] Vieri Benci, Some critical point theorems and applications, Comm. Pure Appl. Math. 33 (1980), no. 2, 147–172. MR 562548, 10.1002/cpa.3160330204
  • [6] Vieri Benci, A geometrical index for the group 𝑆¹ and some applications to the study of periodic solutions of ordinary differential equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 393–432. MR 615624, 10.1002/cpa.3160340402
  • [7] Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, 10.1007/BF01389883
  • [8] David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65–74. MR 0296777
  • [9] Ivar Ekeland and Jean-Michel Lasry, Sur le nombre de points critiques de fonctions invariantes par des groupes, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 11, Ai, A559–A562 (French, with English summary). MR 0415671
  • [10] Edward R. Fadell and Paul H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), no. 2, 139–174. MR 0478189
  • [11] M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0159197
  • [12] L. A. Ljusternik and L. Schnirelmann, Topological methods in the calculus of variations, Hermann, Paris, 1934.
  • [13] 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. MR 0464299
  • [14] Paul H. Rabinowitz, On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 33 (1980), no. 5, 609–633. MR 586414, 10.1002/cpa.3160330504
  • [15] -, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 225-251.
  • [16] -, Variational methods for finding periodic solutions of differential equations, Nonlinear Evolution Equations (M. Crandall, editor), Academic Press, New York, 1978, pp. 225-251.
  • [17] Martin Schechter, Spectra of partial differential operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR 869254
  • [18] H. Weyl, The theory of groups and quantum mechanics, Dover, New York, 1950.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0675067-X
Keywords: Critical point, symmetry, minimax methods, Hamiltonian systems, periodic solutions
Article copyright: © Copyright 1982 American Mathematical Society