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Multiplicity of homoclinic solutions for singular second-order conservative systems

Published online by Cambridge University Press:  14 November 2011

Maria Letizia Bertotti  and
Louis Jeanjean
Maria Letizia Bertotti
Affiliation:
Laboratorio di Matematica Applicata, Ingegneria, Università degli Studi di Trento, 38050 Trento, Italy e-mail: MariaLetizia.Bertotti@ing.unitn.it
Louis Jeanjean
Affiliation:
Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, 93166 Noisy le Grand Cedex, France e-mail: Jeanjean@math.univ.mlv.fr
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Abstract

We consider a class of second-order systems , with q(t) ∊ℝn, for which the potential energy V: ℝn\S→ℝ admits a (possibly unbounded) singular set S ⊂ℝn and has a unique absolute maximum at 0 ∈ℝn. Under some conditions on S and V, we prove the existence of several solutions homoclinic to 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1169 - 1180
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Ambrosetti, A. and Bertotti, M. L.. Homoclinics for second order conservative systems. In Partial Differential Equations and Related Subjects, ed. Miranda, M., Research Notes in Mathematics 269, 21–37 (Harlow: Longman Scientific and Technical, 1992).Google Scholar
2Ambrosetti, A. and Zelati, V. Coti. Periodic Solutions of Singular Lagrangian Systems (Boston: Birkhauser, 1993).CrossRefGoogle Scholar
3Bahri, A. and Rabinowitz, P. H.. A minimax method for a class of Hamiltonian systems with singular potential. J. Fund. Anal. 82 (1989), 412–28.CrossRefGoogle Scholar
4Bertotti, M. L. and Bolotin, S. V.. Homoclinic solutions of quasiperiodic Lagrangian systems. Differential Integral Equations 8 (1995), 1733–60.CrossRefGoogle Scholar
5Bertotti, M. L. and Bolotin, S. V.. A variational approach for homoclinics in almost periodic Hamiltonian systems. Comm. Appl. Nonlinear Anal. 2 (1995), 43–57.Google Scholar
6Bessi, U.. Multiple homoclinic orbits for autonomous, singular potentials. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 785–802.CrossRefGoogle Scholar
7Bolotin, S. V.. Libration motions of reversible Hamiltonian systems (Dissertation, Moscow State University, Moscow, USSR, 1981 (in Russian)).Google Scholar
8Bolotin, S. V.. Homoclinic orbits to invariant tori of Hamiltonian systems. Amer. Math. Soc. Transl. (2) 168 (1995), 21–90. See also S. V. Bolotin. Homoclinic orbits to minimal invariant tori of Lagrangian systems. Vestnik Moskov Univ. Ser. 1 Mat. Mekh, 6 (1992), 34–41 (in Russian).Google Scholar
9Bolotin, S. V. and Kozlov, V. V.. Libration in systems with many degrees of freedom. Prikl. Mat. Mekh. 42 (1978), 245–50; English transl. J. Appl. Math. Mech. 42 (1978), 256–61.Google Scholar
10Caldiroli, P.. Existence and multiplicity of homoclinic orbits for singular potentials on unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 317–39.CrossRefGoogle Scholar
11Caldiroli, P. and Nolasco, M.. Multiple homoclinic solutions for a class of autonomous singular systems in ℝ (Preprint, SISSA, Trieste, 1995).Google Scholar
12Gordon, W. B.. Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204 (1975), 113–35.CrossRefGoogle Scholar
13Massey, W. S.. Algebraic Topology: an Introduction (New York: Springer, 1967).Google Scholar
14Rabinowitz, P. H.. Periodic and heteroclinic orbits for a periodic Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 331–46.CrossRefGoogle Scholar
15Rabinowitz, P. H.. Homoclinics for a singular Hamiltonian system in ℝn (Preprint, Madison, 1994).Google Scholar
16Tanaka, K.. Homoclinic orbits for a singular second order Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 427–38.CrossRefGoogle Scholar