Skip to main content
SpringerLink
Log in

Solutions of mountain pass type for double well potential systems

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

This paper studies a Hamiltonian system possessing a double well potential for which the existence of multitransition heteroclinic and homoclinic solutions that are local minimizers of an associated functional is known. Under an additional mild non-degeneracy condition on the set of all homoclinic and heteroclinic solutions, the existence of further heteroclinic and homoclinic solutions that are of mountain pass type is established. A key tool for the existence arguments is a variant of the Mountain Pass Theorem that is of independent interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bolotin, S.V.: Libration motions of natural dynamical systems. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6, 72–77 (1978). (Russian–English summary)

    MATH  Google Scholar 

  2. Bolotin, S.V.: Existence of homoclinic motions. Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. 6, 98–103 (1983). (Russian)

    MathSciNet  MATH  Google Scholar 

  3. Kozlov, V.V.: Calculus of variations in the large and classical mechanics. Russ. Math. Surv. 40, 37–71 (1985)

    Article  MathSciNet  Google Scholar 

  4. Coti Zelati, V., Ekeland, I., Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288, 133–160 (1990)

    Article  MathSciNet  Google Scholar 

  5. Sternberg, P.: Vector valued local minimizers of nonconvex variational problems. Rocky Mt. J. Math. 21, 799–807 (1991)

    Article  MathSciNet  Google Scholar 

  6. Mather, J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble) 43, 1349–1386 (1993)

    Article  MathSciNet  Google Scholar 

  7. Rabinowitz, P.H.: Homoclinic and heteroclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 1, 1–36 (1993)

    Article  MathSciNet  Google Scholar 

  8. Alikakos, N.D., Fusco, G.: On the connection problem for potentials with several global minima. Indiana Univ. Math. J. 57, 1871–1906 (2008)

    Article  MathSciNet  Google Scholar 

  9. Byeon, J., Montecchiari, P., Rabinowitz, P.H.: A double well potential system. APDE 9, 1737–1772 (2016)

    Article  MathSciNet  Google Scholar 

  10. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 9, 349–381 (1973)

    Article  MathSciNet  Google Scholar 

  11. Bisgard, J.: Homoclinic and Heteroclinic Connections for Two Classes of Hamiltonian Systems. University of Wisconsin-Madison, Doctoral thesis (2005)

  12. Alessio, F., Jeanjean, L., Montecchiari, P.: Stationary layered solutions in \(\mathbb{R}^{2}\) for a class of non autonomous Allen–Cahn equations. Calc. Var. Partial Differ. Equ. 11, 177–202 (2000)

    Article  Google Scholar 

  13. Rabinowitz, P.H., Stredulinsky, E.: Extensions of Moser–Bangert Theory: Locally Minimal Solutions, Progress in Nonlinear Differential Equations and Their Applications, vol. 81. Birkhauser, Boston (2011)

    Book  Google Scholar 

  14. Moser, J.: Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non Lineaire 3, 229–272 (1986)

    Article  MathSciNet  Google Scholar 

  15. Bolotin, S., Rabinowitz, P.H.: A note on heteroclinic solutions of mountain pass type for a class of nonlinear elliptic PDE’s. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 66, pp. 105–114. Birkhauser, Basel (2006)

  16. Bolotin, S., Rabinowitz, P.H.: Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs. Ann. Inst. H. Poincaré Anal. Non Lineaire 31, 103–128 (2014)

    Article  MathSciNet  Google Scholar 

  17. Rabinowitz, P.H.: Connecting orbits for a reversible Hamiltonian system. Ergod. Theory Dyn. Sys. 20, 1767–1784 (2000)

    Article  MathSciNet  Google Scholar 

  18. Rabinowitz, P.H.: On a class of reversible elliptic systems. Netw. Heterog. Media 7, 927–939 (2012)

    Article  MathSciNet  Google Scholar 

  19. Coti Zelati, V., Rabinowitz, P.H.: Homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials. JAMS 4, 693–727 (1991)

    MathSciNet  MATH  Google Scholar 

  20. Whyburn, G.T.: Topological Analysis. Princeton University Press, Princeton (1964)

    Book  Google Scholar 

  21. Montecchiari, P., Rabinowitz, P.H.: On the existence of multi-transition solutions for a class of elliptic systems. Ann. Inst. H. Poincaré Anal. Non Lineaire 33, 199–219 (2016)

    Article  MathSciNet  Google Scholar 

  22. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations CBMS Regional Conference Series in Mathematics, vol. 65. AMS (1986)

  23. Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27–42 (1992)

    Article  MathSciNet  Google Scholar 

  24. Séré, E.: Looking for the Bernoulli shift. Ann. IHP Anal. Nonlinéaire 10, 561–590 (1993)

    Article  MathSciNet  Google Scholar 

  25. Montecchiari, P., Nolasco, M., Terracini, S.: Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems. Calc. Var. PDE 5, 523–555 (1997)

    Article  MathSciNet  Google Scholar 

  26. Hofer, H.: A geometric description of the neighbourhood of a critical point given by the Mountain Pass Theorem. J. Lond. Math. Soc. 31, 566–570 (1985)

    Article  MathSciNet  Google Scholar 

  27. Pucci, P., Serrin, J.: The structure of the critical set in the Mountain Pass Theorem. Trans. AMS 299, 115–132 (1987)

    Article  MathSciNet  Google Scholar 

  28. Pucci, P., Serrin, J.: A mountain pass theorem. J. Differ. Equ. 60, 142–149 (1985)

    Article  MathSciNet  Google Scholar 

  29. Pucci, P., Serrin, J.: Extensions of the Mountain Pass Theorem. J. Funct. Anal. 59, 185–210 (1984)

    Article  MathSciNet  Google Scholar 

  30. Caldiroli, P., Montecchiari, P.: Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Commun. Appl. Nonlinear Anal. 1, 97–129 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via brecce bianche, 60131, Ancona, Italy

    Piero Montecchiari

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Paul H. Rabinowitz

Authors
  1. Piero Montecchiari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul H. Rabinowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul H. Rabinowitz.

Additional information

Communicated by A. Malchiodi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Montecchiari, P., Rabinowitz, P.H. Solutions of mountain pass type for double well potential systems. Calc. Var. 57, 114 (2018). https://doi.org/10.1007/s00526-018-1400-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-018-1400-4

Mathematics Subject Classification

Search

Navigation