Skip to main content
SpringerLink
Log in

Ground state periodic solutions of second order Hamiltonian systems without spectrum 0

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the second order Hamiltonian system

$\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$

Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= −d 2/dt 2 −A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.

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. F. Antonacci, Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign, Nonlinear Analysis 29 (1997), 1353–1364.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Bartsch and M. Willem, Periodic solutions of nonautonomous Hamiltonian systems with symmetries, Journal für die Reine und Angewandte Mathematik 451 (1994), 149–159.

    MathSciNet  MATH  Google Scholar 

  3. V. Benci, Some critical point theorems and applications, Communications on Pure and Applied Mathematics 33 (1980), 147–172.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Chen and S. Ma, Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0, Journal of Mathematical Analysis and Applications 379 (2011), 842–851.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems, Proceedings of the American Mathematical Society 139 (2011), 3973–3983.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Applied Mathematics and Computation 218 (2012), 5496–5507.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Ding and C. Lee, Periodic solutions of Hamiltonian systems, SIAM Journal on Mathematical Analysis 32 (2000), 555–571.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic Journal of Differential Equations 2002 (2002), 1–12.

    Google Scholar 

  9. M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian systems, Nonlinear Analysis 7 (1983), 475–482.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Girardi and M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Analysis 10 (1986), 371–382.

    Article  MathSciNet  MATH  Google Scholar 

  11. Z. Guo and J. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Science in China. Series A. Mathematics 46 (2003), 506–515.

    MathSciNet  MATH  Google Scholar 

  12. X. He and X. Wu, Periodic solutions for a class of nonautonomous second order Hamiltonian systems, Journal of Mathematical Analysis and Applications 341 (2008), 1354–1364.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on R N, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 129 (1999), 787–809.

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. M. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Transactions of American Mathematical Society 311 (1989), 749–780.

    Article  MATH  Google Scholar 

  15. S. J. Li and M. Willem, Applications of local linking to critical point theory, Journal of Mathematical Analysis and Applications 189 (1995), 6–32.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Luan and A. Mao, Periodic solutions for a class of nonautonomous Hamiltonian systems, Nonlinear Analysis 61 (2005), 1413–1426.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Communications on Pure and Applied Mathematics 31 (1978), 157–184.

    Article  MathSciNet  Google Scholar 

  18. P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Transactions of the American Mathematical Society 272 (1982), 753–769.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM. Control, Optimisation and Calculus of Variations 9 (2003), 601–619.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Schechter, Superlinear Schrödinger operators, Journal of Functional Analysis 262 (2012), 2677–2694.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2000.

    Book  MATH  Google Scholar 

  22. A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, Journal of Functional Analysis 187 (2001), 25–41.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, Journal of Functional Analysis 257 (2009), 3802–3822.

    Article  MathSciNet  MATH  Google Scholar 

  24. Z.-L. Tao, S. Yan and S.-L. Wu, Periodic solutions for a class of superquadratic Hamiltonian systems, Journal of Mathematical Analysis and Applications 331 (2007), 152–158.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.

    Book  MATH  Google Scholar 

  26. M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana University Mathematics Journal 52 (2003), 109–132.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Yang, W. Chen and Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum 0, Acta Applicandae Mathematicae 110 (2010), 1475–1488.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Yang, W. Chen and Y. Ding, Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities, Journal of Mathematical Analysis and Applications 364 (2010), 404–413.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Analysis 72 (2010), 2620–2627.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Anyang Normal University, Anyang, 455000, Henan Province, P.R. China

    Guanwei Chen

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Shiwang Ma

Authors
  1. Guanwei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiwang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guanwei Chen.

Additional information

Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, G., Ma, S. Ground state periodic solutions of second order Hamiltonian systems without spectrum 0. Isr. J. Math. 198, 111–127 (2013). https://doi.org/10.1007/s11856-013-0016-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-013-0016-9

Keywords

Search

Navigation