Acta Applicandae Mathematicae

, Volume 110, Issue 3, pp 1353–1371 | Cite as

Homoclinic Orbits of Nonperiodic Super Quadratic Hamiltonian System

  • Jian Ding
  • Junxiang Xu
  • Fubao Zhang


This paper is concerned with solutions of the Hamiltonian system: \(\dot{u}=\mathcal{J}H_{u}(t,u)\) , where \(H(t,u)=\frac{1}{2}u\cdot Lu+W(t,u)\) with L being a 2N×2N symmetric matrix and WC 1(ℝ×ℝ2N ,ℝ) being super quadratic at infinity in u∈ℝ2N . We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Hamiltonian system, we constructed linking levels of the variational functional Φ such that the minimax value E l based on the linking structure of Φ satisfies \(0<E_{l}<E_{l_{0}}\) , where \(E_{l_{0}}\) is the least action of the “limit equation”. Thus we can show the (PS) c -condition holds true for all \(c<E_{l_{0}}\) and consequently we obtain one solution of the Hamiltonian system.


Hamiltonian system Super quadratic Least action solution (PS)c-condition Lions’ concentration compactness principle 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ding, Y.H.: Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms. Commun. Contemp. Math. 8, 453–480 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ding, Y.H., Willem, M.: Homoclinic orbits of a Hamiltonian system. Z. Angew. Math. Phys. 50, 759–778 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ding, Y.H., Girardi, M.: Infinitely many homoclinic orbits of a Hamiltonian system with symmetry. Nonlinear Anal. 38, 391–415 (1999) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ding, Y.H.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25(11), 1095–1113 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ding, Y.H., Li, S.J.: Homoclinic orbits for first order Hamiltonian systems. J. Math. Anal. Appl. 189, 585–601 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ding, Y.H., Jeanjean, L.: Homoclinic orbits for a nonperiodic Hamiltonian system. J. Differ. Equ. 237, 473–490 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bartsch, T., Ding, Y.H.: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nachr. 279, 1267–1288 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ding, Y.H., Lee, C.: Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system. J. Differ. Equ. 246, 2829–2848 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Coti-Zelati, V., Ekeland, I., Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 228, 133–160 (1990) CrossRefGoogle Scholar
  10. 10.
    Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 228, 483–503 (1990) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27–42 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Tanaka, K.: Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits. J. Differ. Equ. 94, 315–339 (1991) zbMATHCrossRefGoogle Scholar
  13. 13.
    Arioli, G., Szulkin, A.: Homoclinic solutions of Hamiltonian systems with symmetry. J. Differ. Equ. 158, 291–313 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Séré, E.: Looking for the Bernoulli shift. Ann. Inst. H. Poincaré Anal. Non Linéaire. 10, 561–590 (1993) zbMATHGoogle Scholar
  15. 15.
    Szulkin, A., Zou, W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 3, 441–472 (1998) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Ackermann, N.: A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. J. Funct. Anal. 234, 423–443 (2006) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingPeople’s Republic of China

Personalised recommendations