Skip to main content
    • Aa
    • Aa

Existence and multiplicity of homoclinic orbits for potentials on unbounded domains

  • Paolo Caldiroli (a1)

We study the system in RN, where V is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimising argument, we can prove the existence of a homoclinic orbit when the component Ω of {x ∈ RN: V(x) < V(0)} containing 0 is an arbitrary open set; in the case Ω unbounded we allow V(x) to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in Ω implies that a homoclinic solution can also be found via a minimax procedure and, comparing the critical levels of the functional associated to the system, we see that the two solutions are distinct whenever the singularity is ‘not too far’ from 0.

Hide All
1Ambrosetti A. and Bertotti M. L.. Homoclinics for second order conservative systems. In Partial Differential Equations and Related Subjects, ed. Miranda M., Pitman Research Notes in Math. Ser. (London: Pitman Press, 1992).
2Ambrosetti A. and Coti Zelati V.. Multiple homoclinic orbits for a class of conservative systems. Rend. Sent. Univ. Padova 89 (1993), 177194.
3Bahri A. and Rabinowitz P. H.. A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 82 (1989), 412428.
4Benci V. and Giannoni F.. Homoclinic orbits on compact manifolds. J. Math. Anal. Appl. 157 (1991), 568576.
5Coti Zelati V., Ekeland I. and Séré E.. A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 133160.
6Coti Zelati V. and Rabinowitz P. H.. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991), 693727.
7Coti Zelati V. and Serra E.. Multiple brake orbits for some classes of singular Hamiltonian systems (Preprint, 1991).
8Coti V. Zelati and Serra E.. Collision and non-collision solutions for a class of Keplerian-like dynamical systems (Preprint, SISSA, 1991).
9Gordon W. B.. Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204 (1975), 113135.
10Hofer H. and Wysocki K.. First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 483503.
11Lions P. L.. The concentration-compactness principle in the calculus of variations. Rev. Mat. Iberoamericana 1 (1985), 145201.
12Poincaré H.. Les Methodes Nouvelles de la Méchanique Céleste (Paris: Gauthier–Villars, 18971899).
13Rabinowitz P. H.. Periodic and heteroclinic orbits for a periodic Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (5) (1989), 331346.
14Rabinowitz P. H.. Homoclinic orbits for a class of Hamiltonian systems. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 3338.
15Rabinowitz P. H. and Tanaka K.. Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206 (1991), 473499.
16Séré E.. Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209 (1992), 2742.
17Séré E.. Homoclinic orbits on compact hypersurfaces in R2N, of restricted contact type (Preprint, CEREMADE, 1992).
18Séré E.. Looking for the Bernoulli shift. Ann. Inst. H. Poincaré Anal. Non Lineaire 10 (5) (1993), 561590.
19Tanaka K.. Homoclinic orbits for a singular second order Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (5) (1990), 427438.
20Tanaka K.. Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits. J. Differential Equations 94 (1991), 315339.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 34 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.