Advertisement

Ground State Homoclinic Orbits for First-Order Hamiltonian System

  • Wen Zhang
  • Jian ZhangEmail author
  • Xianhua Tang
Article
  • 11 Downloads

Abstract

In this paper, we study the following first-order Hamiltonian system
$$\begin{aligned} \dot{z}=\mathscr {J}H_{z}(t,z). \end{aligned}$$
Under some suitable conditions on the nonlinearity, we establish the existence of ground state homoclinic orbits by using variational methods. Moreover, we also explore some properties of these homoclinic orbits, such as compactness of set of and exponential decay of ground state homoclinic orbits.

Keywords

Hamiltonian system Ground state homoclinic orbits Exponential decay Variational methods 

Mathematics Subject Classification

37J45 70H05 

Notes

References

  1. 1.
    Bartsch, T., Szulkin, A.: Hamiltonian systems: periodic and homoclinic solutions by variational methods. In: Handbook of Differential Equations: Ordinary Differential Equations, vol. II, Elsevier B. V., Amsterdam, pp. 77–146 (2005)Google Scholar
  2. 2.
    Coti-Zelati, V., Ekeland, I., Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 228, 133–160 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coti-Zelati, V., Rabinowitz, P.: Homoclinic orbits for second order Hamiltonian systems possessing superqudratic potentials. J. Am. Math. Soc. 4, 693–727 (1991)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, G.W., Ma, S.W.: Homoclinic orbits of superquadratic Hamiltonian system. Proc. Am. Math. Soc. 139, 3973–3983 (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ding, Y.H.: Variational Methods for Strongly Indefinite Problems. World Scientific Press, Singapore (2008)Google Scholar
  6. 6.
    Ding, Y.H.: Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms. Commun. Contemp. Math. 4, 453–480 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ding, Y.H., Girardi, M.: Infinitely many homoclinic orbits of a Hamiltonian system with symmetry. Nonlinear Anal. 38, 391–415 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ding, Y.H., Jeanjean, L.: Homoclinic orbits for nonperiodic Hamiltonian system. J. Differ. Equ. 237, 473–490 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ding, Y.H., Li, S.J.: Homoclinic orbits for first order Hamiltonian systems. J. Math. Anal. Appl. 189, 585–601 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, Y.H., Lee, C.: Existence and exponential decay of homoclinics in a nonperiodic superquadrtic Hamiltonian system. J. Differ. Equ. 246, 2829–2848 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Ding, Y.H., Willem, M.: Homoclinic orbits of a Hamiltonian system. Z. Angew. Math. Phys. 50, 759–778 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Esteban, M.J., Séré, E.: Stationary states of nonlinear Dirac equations: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hofer, H., Wysocki, K.: First order ellipic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 228, 483–503 (1990)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kryszewki, W., Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 3, 441–472 (1998)zbMATHGoogle Scholar
  15. 15.
    Lions, P.L.: The concentration compactness principle in the calculus of variations. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Nonlinéaire 1, 223–283 (1984)CrossRefzbMATHGoogle Scholar
  16. 16.
    Li, G.B., Szulkin, A.: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 4, 763–776 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  18. 18.
    Omana, W., Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Differ. Int. 5, 1115–1120 (1992)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27–42 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447–526 (1982)CrossRefzbMATHGoogle Scholar
  22. 22.
    Szulkin, A., Zou, W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sun, J., Chu, J., Feng, Z.: Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete Contin. Dyn. Syst. 33, 3807–3824 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802–3822 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Silva, E.A., Vieira, G.F.: Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. PDE 39, 1–33 (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Tanaka, K.: Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits. J. Differ. Equ. 94, 315–339 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tang, X.H.: Non-Nehari manifold method for superlinear Schrödinger equation. Taiwan J. Math. 18, 1957–1979 (2014)CrossRefzbMATHGoogle Scholar
  28. 28.
    Tang, X.H.: Non-Nehari manifold method for asymptotically periodic Schrödinger equations. Sci. China Math. 58, 715–728 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Willem, M.: Minimax Theorems. Birkhäuser, Berlin (1996)CrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, J., Zhang, H., Xu, J., Zhang, F.: Existence of infinitely many homoclinic orbits for nonperiodic superquadratic Hamiltonian systems. Nonlinear Anal. 75, 4873–4883 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang, Q., Liu, C.: Homoclinic orbits for a class of first order nonperiodic Hamiltonian systems. Nonlinear Anal. RWA 41, 34–52 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhang, W., Zhang, J., Mi, H.: On fractional Schrödinger equation with periodic and asymptotically periodic conditions. Comput. Math. Appl. 74, 1321–1332 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, J., Tang, X., Zhang, W.: Homoclinic orbits of nonperiodic superquadratic Hamiltonian system. Taiwan. J. Math. 17, 1855–1867 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zhang, W., Tang, X., Zhang, J.: Homoclinlic solutions for the first-order Hamiltonian system with superquadratic nonlinearity. Taiwan. J. Math. 19, 673–690 (2015)CrossRefzbMATHGoogle Scholar
  35. 35.
    Zhang, J., Zhang, W., Tang, X.: Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete Contin. Dyn. Syst. 37, 4565–4583 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, J., Zhang, W., Zhao, F.: Existence and exponential decay of ground-state solutions for a nonlinear Dirac equation. Z. Angew. Math. Phys. 69, 116 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHunan University of CommerceChangshaPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

Personalised recommendations