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Poincaré–Birkhoff results for Lagrangian systems with subquadratic potential

Published online by Cambridge University Press:  14 November 2011

Silvia Cingolani  and
Lorenzo Pisani
Silvia Cingolani
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bari, Campus Universitario, via E. Orabona, 4, 70125 Bari, Italy
Lorenzo Pisani
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bari, Campus Universitario, via E. Orabona, 4, 70125 Bari, Italy
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Abstract

In this paper, using a recent generalisation of Morse Theory, we study the existence of periodic solutions of the Lagrangian equation (1.1) with subquadratic potential and asymptotically flat, nonconstant, time-dependent metric on ℝN. In Section 3, we get also an ‘alternative result’ about the minimal period or the existence of infinitely many solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1169 - 1177
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Amann, H. and Zehnder, E.. Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm Sup. Pisa 7 (1980), 539–603.Google Scholar
2Benci, V.. Introduction to Morse theory–a new approach. In Proc. Topological Nonlinear Analysis: Degree, Singularity and variations, eds Matzeu, M. and Vignvoli, A., 37–177. (Boston: Birkhäuser, 1995).CrossRefGoogle Scholar
3Benci, V. and Fortunate, D.Periodic solutions of asymptotically linear dynamical systems. Nonlinear Differential Equations Appl. 1 (1994) 267–80.CrossRefGoogle Scholar
4Benci, V. and Giannoni, F.. Morse theory for C1 functionals and Conley blocks (Preprint, 1st. U. Dini, Univ. Pisa, 1993).Google Scholar
5Capozzi, A., Fortunato, D. and Salvatore, A.. Periodic solutions of Lagrangian systems with a bounded potential. J. Math. Anal. Appl. 124 (1987), 482–94.CrossRefGoogle Scholar
6Chang, K. C.. Infinite dimensional Morse theory and multiple solution problems (Boston: Birkhäuser, 1993).CrossRefGoogle Scholar
7Conley, C. and Zehnder, E.. Morse type index theory for flow and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. 37 (1984), 207–53.CrossRefGoogle Scholar
8Ekeland, I.. Convexity Methods in Hamiltonian Mechanics (New York: Springer, 1990).CrossRefGoogle Scholar
9Masiello, A. and Pisani, L.. Asymptotically linear elliptic problems at resonance. Annali Mat. Pura Appl. (to appear).Google Scholar
10Mawhin, J. and Willem, M.. Critical Point Theory and Hamiltonian Systems (New York: Springer, 1989).CrossRefGoogle Scholar
11Mirenghi, E. and Tucci, M.. Existence of Aperiodic solutions for a class of Lagrangian system. Rend. Sem. Mat. Univ. Padova 83 (1990), 19–32.Google Scholar