Advertisement

Journal of Dynamical and Control Systems

, Volume 25, Issue 2, pp 159–174 | Cite as

Infinitely Many Rotating Periodic Solutions for Second-Order Hamiltonian Systems

  • Guanggang Liu
  • Yong LiEmail author
  • Xue Yang
Article
  • 138 Downloads

Abstract

In this paper, we consider a class of second-order Hamiltonian system in \(\mathbb {R}^{N}\) with combined nonlinearities. We will study the multiplicity of rotating periodic solutions, i.e., \(x(t+T)=Qx(t)\) with \(T>0\) and Q is an \(N\times N\) orthogonal matrix. In the case \(Q^{k}\neq I_{N}\) for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution; In the case \(Q^{k}=I_{N}\) for some positive integer k, such a rotating periodic solution is just a subharmonic solution. We will use the Fountain Theorem and its dual form to obtain two sequences of rotating periodic solutions with the corresponding energy tending to infinity and zero respectively.

Keywords

Second-order Hamiltonian systems Rotating periodic solutions Fountain theorem 

Mathematics Subject Classification 2010

34C25 37B30 37J45 

Notes

Acknowledgements

The authors are grateful to the reviewer for careful reading of the paper and helpful suggestions.

References

  1. 1.
    Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J Funct Anal 1973;14(4):349–381.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal 1994;122(2):519–543.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bahri A, Berestycki H. Existence of forced oscillations for some nonlinear differential equations. Commun Pur Appl Math 1984;37(4):403–442.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bartsch T, Willem M. On an elliptic equation with concave and convex nonlinearities. Proc Am Math Soc 1995;123(11):3555–3561.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bartsch T, Clapp M. Critical point theory for indefinite functionals with symmetries. J Funct Anal 1996;138(1):107–136.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bartsch T, de Figueiredo DG. Infinitely many solutions of nonlinear elliptic systems. Topics in nonlinear analysis. Basel: Birkhäuser; 1999, pp. 51–67.zbMATHGoogle Scholar
  7. 7.
    Benci V, Fortunato D. Periodic solutions of asymptotically linear dynamical systems. Nonlinear Differ Equ Appl 1994;1(1):267–280.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cerami G. An existence criterion for the critical points on unbounded manifolds. Istit Lombardo Accad Sci Lett Rend A 1978;112(2):332–336.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chang X, Li Y. Rotating periodic solutions of second order dissipative dynamical systems. Discrete Cont Dyn Syst 2016;36(2):643–652.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chang X, Li Y. Rotating periodic solutions for second-order dynamical systems with singularities of repulsive type. Math Methods Appl Sci 2017;40(8):3092–3099.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chen G, Long Y. Periodic solutions of second order nonlinear Hamiltonian systems with superquadratic potentials having zero mean value. Chinese Annal Math Ser A 1998;219(4):323–341.zbMATHGoogle Scholar
  12. 12.
    Ding Y, Li S. A remark on periodic solutions of singular hamiltonian systems with sublinear terms. Syst Sci Math Sci 1992;5(2):121–126.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hu X, Wang P. Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems. J Funct Anal 2011;261(11):3247–3278.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hu X, Ou Y, Wang P. Trace formula for Linear Hamiltonian systems with its applications to elliptic Lagrangian solutions. Arch Ration Mech Anal 2015;216 (1):313–357.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Jeanjean L. On the existence of bounded palaisCSmale sequences and application to a landesmanCLazer-type problem set on \(\mathbb {R}^{N}\). Proc Roy Soc Edinb 1999;129(4):787–809.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lazzo M. Nonlinear differential problems and Morse theory. Nonlinear Anal 1997; 30(30):169–176.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Liu JQ, Li S. An existence theorem for multiple critical points and its application. Kexue Tongbao 1984;17:1025–1027.MathSciNetGoogle Scholar
  18. 18.
    Liu GG, Li Y, Yang X. Roatting periodic solutions for asymptotically linear second-order Hamiltonian systems with resonance at infinity. Math Methods Appl Sci 2017;40(18):7139–7150.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Liu GG, Li Y, Yang X. Rotating periodic solutions for super-linear second order Hamiltonian systems. Appl Math Lett 2018;79:73–79.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu GG, Li Y, Yang X. Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems. Journal of Differential Equations.  https://doi.org/10.1016/j.jde.2018.04.001.
  21. 21.
    Liu S, Li S. An elliptic equation with concave and convex nonlinearities. Nonlinear Anal Theory Methods Appl 2003;53(6):723–731.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Liu Z, Wang ZQ. On Clark’s theorem and its applications to partially sublinear problems. Annales De Linstitut Henri Poincare Non Linear Analysis 2014;32(5):1015–1037.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Long Y. Multiple solutions of perturbed superquadratic second order hamiltonian systems. Trans Am Math Soc 1989;311(2):749–780.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Long Y. Periodic solutions of superquadratic Hamiltonian systems with bounded forcing terms. Math Z 1990;203(1):453–467.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Long Y. Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials. Nonlinear Anal 1995;24(12):1665–1671.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ma S. Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems. J Differ Equ 2010;248(10):2435–2457.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mawhin J, Willem M. Critical point theory and hamiltonian systems. New York: Springer; 1989.zbMATHCrossRefGoogle Scholar
  28. 28.
    Motreanu D, Motreanu VV, Papageorgiou NS. Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential. Topol Methods Nonlinear Anal 2004;24:269–296.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pipan J, Schechter M. Non-autonomous second order Hamiltonian systems. J Differ Equ 2014;257(2):351–373.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Rabinowitz PH. On subharmonic solutions of hamiltonian systems. Commun Pur Appl Math 1980;33(5):609–633.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Tang CL. Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc Am Math Soc 1998;126(11):3263–3270.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Tang CL, Wu XP. Periodic solutions for a class of new superquadratic second order Hamiltonian systems. Appl Math Letters 2014;34(2):65–71.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Tang XH, Meng Q. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal Real World Appl 2010;11(5):3722–3733.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Wu T. On semilinear elliptic equations involving concaveCconvex nonlinearities and sign-changing weight function. J Math Anal Appl 2006;318(1):253–270.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Zou W. Variant fountain theorems and their applications. Manuscripta Mathematica 2001;104(3):343–358.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Zou W, Li S. Infinitely many solutions for Hamiltonian systems. J Differ Equ 2002;186(1):141–164.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesLiaocheng UniversityLiaochengPeople’s Republic of China
  2. 2.School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary SciencesNortheast Normal UniversityChangchunPeople’s Republic of China
  3. 3.College of MathematicsJilin UniversityChangchunPeople’s Republic of China

Personalised recommendations