Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 4
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Costa, David G. and Tehrani, Hossein 2014. On a class of singular second-order Hamiltonian systems with infinitely many homoclinic solutions. Journal of Mathematical Analysis and Applications, Vol. 412, Issue. 1, p. 200.

    Zhang, Ziheng Liao, Fang-Fang and Wong, Patricia J. Y. 2014. Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems. Abstract and Applied Analysis, Vol. 2014, p. 1.

    Rabinowitz, Paul H. 1997. A multibump construction in a degenerate setting. Calculus of Variations and Partial Differential Equations, Vol. 5, Issue. 2, p. 159.

    Bertotti, Maria Letizia and Jeanjean, Louis 1996. Multiplicity of homoclinic solutions for singular second-order conservative systems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 126, Issue. 06, p. 1169.

  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 124, Issue 2
  • January 1994, pp. 317-339

Existence and multiplicity of homoclinic orbits for potentials on unbounded domains

  • Paolo Caldiroli (a1)
  • DOI:
  • Published online: 14 November 2011

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

3A. Bahri and P. H. Rabinowitz . A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 82 (1989), 412428.

4V. Benci and F. Giannoni . Homoclinic orbits on compact manifolds. J. Math. Anal. Appl. 157 (1991), 568576.

5V. Coti Zelati , I. Ekeland and E. Séré . A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 133160.

6V. Coti Zelati and P. H. Rabinowitz . Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991), 693727.

9W. B. Gordon . Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204 (1975), 113135.

10H. Hofer and K. Wysocki . First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 483503.

15P. H. Rabinowitz and K. Tanaka . Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206 (1991), 473499.

16E. Séré . Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209 (1992), 2742.

20K. Tanaka . 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? *