Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the non-existence of homoclinic orbits
for a class of infinite dimensional
Hamiltonian systems


Authors: Ph. Clément and R. C. A. M. van der Vorst
Journal: Proc. Amer. Math. Soc. 125 (1997), 1167-1176
MSC (1991): Primary 35J50, 35J55, 46E35
DOI: https://doi.org/10.1090/S0002-9939-97-03696-4
MathSciNet review: 1363415
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for a class of infinite dimensional Hamiltonian systems certain homoclinic connections to the origin cease to exist when the non-linearities have `super-critical' growth. The proof is based on a variational principle and a Poho\v{z}aev type identity.


References [Enhancements On Off] (What's this?)

  • 1. Bahri, A. and Coron, J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294. MR 89c:35053
  • 2. Clément, Ph., Felmer, P. and Mitidieri, E., Solutions homoclines d'un système hamiltonien non-borné et superquadratique, C.R. Acad. Sci. Paris 320 (1995), 1481-1484. MR 96f:35072
  • 3. Henry, D., The geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981. MR 83j:35084
  • 4. Hulshof, J and van der Vorst, R.C.A.M., On the equation $\Delta u + u^p=0$, Course Notes Leiden/Delft 1991, `Topics in Nonlinear Analysis'.
  • 5. Mitidieri, E., A Rellich type identity and applictions, Comm. PDE 18 (1993), 125-151. MR 94c:26016
  • 6. Noether, E., Invariante Variations probleme, Nachr. König. Gesell. Wissen. Göttingen, Math.-Phys. Kl. (1918), 235-257.
  • 7. Olver, P.J., Applications of Lie Groups to Differential Equations, Springer-Verlag (GTM), New York, 1986; 2nd ed., 1993. MR 88f:58161; MR 94g:54260
  • 8. Poho\v{z}aev, S.I., Eigenfunctions of the equations $\Delta u + \lambda f(u) =0$, Soviet Math. Dokl. 6 (1965), 1408-1411. MR 33:411
  • 9. Pucci, P. and Serrin, J., A general variational indentity, Indiana Univ. Math. J 35 (1986), 681-703. MR 88b:35072
  • 10. Protter, M. H. and Weinberger, H. F., Maximum Principles in Differential Equations, Prentice Hall: Ehgelwood Cliffs, NJ, 1967. MR 38:2935
  • 11. Smoller, J., Shock Waves and Reaction-Diffusion Equations, Grundlehren der Math. Wissensch., 2nd Ed., vol. 258, 1994. MR 95g:35002
  • 12. van der Vorst, R.C.A.M., Variational Identities and applications to Differential Systems, Arch. Rat. Mech. Anal. 116 (1991), 375-398. MR 93d:35043

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35J50, 35J55, 46E35

Retrieve articles in all journals with MSC (1991): 35J50, 35J55, 46E35


Additional Information

Ph. Clément
Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, Delft, The Netherlands

R. C. A. M. van der Vorst
Affiliation: Center for Dynamical Systems, Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332-0190

DOI: https://doi.org/10.1090/S0002-9939-97-03696-4
Received by editor(s): October 25, 1995
Additional Notes: This work was supported by the Netherlands Organization for Scientific Research, NWO and EC-HCM project Reaction–Diffusion Equations ERBCHRXCT930409.
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society