Topological Methods in Nonlinear Analysis

Homoclinic orbits of first order nonlinear Hamiltonian systems with asymptotically linear nonlinearities at infinity

Guanwei Chen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

By using variational methods and critical point theory, in particular, a generalized weak linking theorem, we study a first order nonlinear Hamiltonian system with asymptotically linear nonlinearity at infinity. We obtain the existence of ground state homoclinic orbits for this nonlinear Hamiltonian system. In particular, we obtain a necessary and sufficient condition for the existence of ground state homoclinic orbits. To the best of our knowledge, there is no published result focusing on necessary and sufficient conditions of the existence of ground state homoclinic orbits for this system.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 2 (2016), 499-510.

Dates
First available in Project Euclid: 13 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1468413749

Digital Object Identifier
doi:10.12775/TMNA.2016.038

Mathematical Reviews number (MathSciNet)
MR3559623

Zentralblatt MATH identifier
1361.37060

Citation

Chen, Guanwei. Homoclinic orbits of first order nonlinear Hamiltonian systems with asymptotically linear nonlinearities at infinity. Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 499--510. doi:10.12775/TMNA.2016.038. https://projecteuclid.org/euclid.tmna/1468413749


Export citation

References

  • G. Arioli and A. Szulkin, Homoclinic solutions of Hamiltonian systems with symmetry, J. Differential Equations, 158 (1999), 291–313.
    Mathematical Reviews (MathSciNet): MR1721901
    Digital Object Identifier: doi:10.1006/jdeq.1999.3639
  • G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems, Proc. Amer. Math. Soc. 139 (2011), 3973–3983.
    Mathematical Reviews (MathSciNet): MR2823043
    Digital Object Identifier: doi:10.1090/S0002-9939-2011-11185-7
  • V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1990), 133–160.
    Mathematical Reviews (MathSciNet): MR1070929
    Digital Object Identifier: doi:10.1007/BF01444526
  • Y.H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math. 8 (2006), 453–480.
    Mathematical Reviews (MathSciNet): MR2258874
    Digital Object Identifier: doi:10.1142/S0219199706002192
  • ––––, Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., vol. 7, World Scientific, 2007.
  • Y.H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal. 38 (1999), 391–415.
    Mathematical Reviews (MathSciNet): MR1705761
  • Y.H. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys. 50 (1999), 759–778.
    Mathematical Reviews (MathSciNet): MR1721793
  • H. Hofer and K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 483–503.
    Mathematical Reviews (MathSciNet): MR1079873
    Digital Object Identifier: doi:10.1007/BF01444543
  • P.L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact cases, Parts I and II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.
    Mathematical Reviews (MathSciNet): MR778974
  • E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209 (1992), 27–42.
    Mathematical Reviews (MathSciNet): MR1143210
    Digital Object Identifier: doi:10.1007/BF02570817
  • ––––, Looking for the Bernoulli shift, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 561–590.
    Mathematical Reviews (MathSciNet): MR1249107
  • M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var. 9 (2003), 601–619.
    Mathematical Reviews (MathSciNet): MR1998717
    Digital Object Identifier: doi:10.1051/cocv:2003029
  • K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: convergence of subharmonic orbits, J. Differential Equations 94 (1991), 315–339.
    Mathematical Reviews (MathSciNet): MR1137618
    Digital Object Identifier: doi:10.1016/0022-0396(91)90095-Q
  • M. Willem, Minimax Theorems, Birkhäuser, 1996.
    Mathematical Reviews (MathSciNet): MR1400007
  • X. Xu, Homoclinic orbits for first order hamiltonian systems possessing super-quadratic potentials, Nonlinear Anal. 51 (2002), 197–214.
    Mathematical Reviews (MathSciNet): MR1918340
  • S. Zhang, Symmetrically Homoclinic Orbits for Symmetric Hamiltonian Systems, J. Math. Anal. Appl. 247 (2000), 645–652.
    Mathematical Reviews (MathSciNet): MR1769099
    Digital Object Identifier: doi:10.1006/jmaa.2000.6839

Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies

Topological Methods in Nonlinear Analysis

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more