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Multiple homoclinic orbits for autonomous, singular potentials

Published online by Cambridge University Press:  14 November 2011

Ugo Bessi
Ugo Bessi
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy
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Abstract

We consider the problem

where u ∊ Rn, n ≧ 2, and V ∊ C2(Rn∖e, R) is a potential having an absolute maximum at 0 and such that V(x) → − ∞ as x → e. We prove that, under some conditions on V, this problem has at least n − 1 geometrically distinct solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 4 , 1994 , pp. 785 - 802
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Al'ber, S. I.. On periodicity problems in the calculus of variations in the large. Russian Math. Surveys 25(4) (1970), 51–117.CrossRefGoogle Scholar
2Al'ber, S. I.. Homologies of the space of nonoriented loops and their application to the calculus of variations in the large. Soviet Math. Dokl. 5 (1964), 312–316.Google Scholar
3Bessi, U.. Multiple closed orbits of fixed energy for gravitational potentials. Rend. Sem. Mat. Univ. Padova 85 (1991), 201–215.Google Scholar
4Coti Zelati, V., Ekeland, I. and Seré, E.. A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 133–160.CrossRefGoogle Scholar
5Coti Zelati, V. and Rabinowitz, P. H.. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991), 693–727.CrossRefGoogle Scholar
6Giannoni, F. and Degiovanni, M.. Dynamical systems with Newtonian-type potentials. Ann. Scuola Norm. Sup. Pisa 15 (1988), 467–494.Google Scholar
7Lions, P. L.. La methode de concentration-compacité en calcul des variations. Seminaire Goulaouic–Meyer–Schwartz, 1982–1983.Google Scholar
8Mel'nikov, V. K.. On the stability of the center for time-periodic perturbations. Trans. Moscow Math. Soc. 12 (1963), 1–56.Google Scholar
9Palais, R. S.. Homotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1–16.CrossRefGoogle Scholar
10Rabinowitz, P. H.. Homoclinic orbits for a class of Hamiltonian systems. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 33–38.CrossRefGoogle Scholar
11Tanaka, K.. Homoclinic orbits for a singular second order dynamical system. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(5) (1990), 427–438.CrossRefGoogle Scholar