Skip to main content
SpringerLink
Log in

Multiple solutions for a superlinear and periodic elliptic system on \({\mathbb{R}^N}\)

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper is concerned with the following periodic Hamiltonian elliptic system

$$\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.$$

where the potential V is periodic and 0 lies in a gap of the spectrum of −Δ + V, f(x, t) and g(x, t) depend periodically on x and are superlinear but subcritical in t at infinity. By establishing a variational setting, existence of a ground state solution and multiple solution for odd f and g are obtained.

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. Alves C.O., Carrião P.C., Miyagaki O.H.: On the existence of positive solutions of a perturbed Hamiltonian system in \({\mathbb{R}^N}\). J. Math. Anal. Appl. 276, 673–690 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ávila A.I., Yang J.: On the existence and shape of least energy solutions for some elliptic systems. J. Diff. Eqns. 191, 348–376 (2003)

    Article  MATH  Google Scholar 

  3. Bartsch T., De Figueiredo D.G.: Infinitely Many Solutions of Nonlinear Elliptic Systems, Progress in Nonlinear Differential Equations and Their Applications, vol. 35, pp. 51–67. Birkhäuser, Basel/Switzerland (1999)

    Google Scholar 

  4. Bartsch T., Ding Y.: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nach. 279, 1–22 (2006)

    Article  MathSciNet  Google Scholar 

  5. Benci V., Rabinowitz P.H.: Critical point theorems for indefinite functionals. Inven. Math. 52, 241–273 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coti Zelati V., Rabinowitz P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4, 693–727 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Figueiredo D.G., Ding Y.: Strongly indefinite functionals and multiple solutions of elliptic systems. Tran. Am. Math. Soc. 355, 2973–2989 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Figueiredo D.G., Felmer P.L.: On superquadratic elliptic systems. Tran. Am. Math. Soc. 343, 97–116 (1994)

    MathSciNet  Google Scholar 

  9. De Figueiredo D.G., DO Ó B., Ruf J.M.: An Orlicz-space approach to superlinear elliptic systems. J. Func. Anal. 224, 471–496 (2005)

    Article  MATH  Google Scholar 

  10. De Figueiredo D.G., Yang J.: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 33, 211–234 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gohberg I., Goldberg S., Kaashoek M.A.: Classed of Linear Operators, vol. I. Birkhäuser-Verlag, Basel, Boston, Berlin (1990)

    Google Scholar 

  12. Hulshof J., Van De Vorst R.C.A.M.: Differential systems with strongly variational structure. J. Func. Anal. 114, 32–58 (1993)

    Article  MATH  Google Scholar 

  13. Jeanjean L.: On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on \({\mathbb{R}^N}\). Proc. R. Soc Edinb. 129 A, 787–809 (1999)

    MathSciNet  Google Scholar 

  14. Kato T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin, Heidelberg, New York (1976)

    MATH  Google Scholar 

  15. Kryszewski W., Szulkin A.: An infinite dimensional Morse theory with applications. Tran. Am. Math. Soc. 349, 3181–3234 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li G., Yang J.: Asymptotically linear elliptic systems. Comm. Partial Diff. Eqns. 29, 925–954 (2004)

    Article  MATH  Google Scholar 

  17. Li Y., Wang Z., Zeng J.: Ground states of nonlinear Schrödinger equations with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 829–837 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lions P.L.: The concentration compactness principle in the calculus of variations. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223–283 (1984)

    MATH  Google Scholar 

  19. Pistoia A., Ramos M.: Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Diff. Eqns. 201, 160–176 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Reed M., Simon B.: Methods of Modern Mathematical Physics, IV Analysis of Operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  21. Séré E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 133–160 (1992)

    Article  Google Scholar 

  22. Sirakov B.: On the existence of solutions of Hamiltonian elliptic systems in R N. Adv. Diff. Eqns. 5, 1445–1464 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Sirakov B., Soares S.H.M.: Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type. Trans. Am. Math. Soc. 362, 5729–5744 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Szulkin A., Weth T.: Ground state solutions for some indefinite problems. J. Funct. Anal. 257, 3802–3822 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Szulkin A., Zou W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, North-Holland (1978)

    Google Scholar 

  27. Van der Vorst R.C.A.M.: Variational identities and applications to differential systems. Arch. Ration. Mech. Anal. 116, 375–398 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhao F., Zhao L., Ding Y.: Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. ESAIM Control Optimisation Calc. Var. 16, 77–91 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yunnan Normal University, 650092, Kunming, Yunnan, People’s Republic of China

    Fukun Zhao

  2. Department of Mathematics, Beijing University of Chemical technology, 100029, Beijing, People’s Republic of China

    Leiga Zhao

  3. Institute of Mathematics, AMSS, CAS, 100080, Beijing, People’s Republic of China

    Fukun Zhao & Yanheng Ding

Authors
  1. Fukun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leiga Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanheng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fukun Zhao.

Additional information

Supported partly by NSFC(11061040 and 11001008), NSFY of Yunnan Province(2008CD112 and 2009CD043) and the Foundation of Education Committee of Yunnan Province, China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, F., Zhao, L. & Ding, Y. Multiple solutions for a superlinear and periodic elliptic system on \({\mathbb{R}^N}\) . Z. Angew. Math. Phys. 62, 495–511 (2011). https://doi.org/10.1007/s00033-010-0105-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-010-0105-0

Mathematics Subject Classification (2000)

Keywords

Search

Navigation