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

DOI:
https://doi.org/10.1090/S0002-9947-1982-0675067-X

MathSciNet review:
675067

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

DOI:
https://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