Skip to main content
SpringerLink
Log in

Existence of solutions for a system of diffusion equations with spectrum point zero

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

Abstract

In this paper, we consider the existence and multiplicity of homoclinic type solutions to a system of diffusion equations with spectrum point zero. By using some recent critical point theorems for strongly indefinite problems, we obtain at least one nontrivial solution and also infinitely many solutions.

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. Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  2. Nagasawa M.: Schrödinger Equations and Diffusion Theory. Birkhöser, Basel (1993)

    Book  MATH  Google Scholar 

  3. Barbu V.: Periodic solutions to unbounded Hamiltonian system. Discret. Contin. Dyn. Syst. 1, 277–283 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ding Y., Lee C.: Multiple solutions of Schrödinger equations with indefinite linear part and super or asympototically linear terms. J. Differ. Equ. 222, 137–163 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bartsch T., Ding Y.: Homoclinic solutions of an infinite-dimensional Hamiltonian system. Math. Z. 240, 289–310 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ding Y., Luan S., Willem M.: Solutions of a system of diffusion equations. J. Fixed Point Theory Appl. 2, 117–139 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brézis H., Nirenberg L.: Characterization theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nachr. 279, 1267–1288 (2006)

    Article  MathSciNet  Google Scholar 

  8. Clément P., Felmer P., Mitidieri E.: Solutions homoclines d’un système hamiltonien non-borné et superquadratique. C.R. Acad. Sci. Paris 320, 1481–1484 (1995)

    MATH  Google Scholar 

  9. Clément P., Felmer P., Mitidieri E.: Homoclinic orbits for a class of infinite dimensional Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa 24(2), 367–393 (1997)

    MATH  MathSciNet  Google Scholar 

  10. Bartsch T., Ding Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313, 15–37 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ding, Y.: Variational Methods for Strongly Indefinite Problems. Interdiscip. Math. Sci., vol. 7. World Scientific, Singapore (2007)

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

    Article  MathSciNet  Google Scholar 

  13. Coti-Zelati V., Rabinowitz P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \({\mathbb{R}^N}\). Commun. Pure Appl. Math. 45, 1217–1269 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

  15. Ding Y., Willem M.: Homoclinic orbits of a Hamiltonian system. Z. Angew. Math. Phys. 50, 759–778 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. de Figueiredo, D.G.: Chapter 1: semilinear elliptic systems: existence, multiplicity, symmetry of solutions. Handb. Differ. Equ. Station. Partial Differ. Equ. 5, 1–40 (2008)

  17. de Figueiredo D.G., Ding Y.: Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Am. Math. Soc. 355, 2973ndash;2989 (2003)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Mawhin, J., Willem, M: Critical Point Theory and Hamilatonian Systems. Appl. Math. Sci., vol. 74. Springer (1989)

  21. Schechter M., Zou W.: Homoclinic orbits for Schrödinger systems. Mich. Math. J. 51, 59–71 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Willem M., Zou W.: On a Schrödinger Equation with periodic potential and spectrum point zero. Indiana Univ. Math. J. 52, 109–132 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yang M., Chen W., Ding Y.: Solutions of a class of Hamiltonian elliptic systems in \({\mathbb{R}^N}\). J. Math. Anal. Appl. 362, 338–349 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Yang M., Chen W., Ding Y.: Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities. J. Math. Anal. Appl. 364, 404C413 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhao F., Ding Y.: On Hamiltonian elliptic systems with periodic or non-periodic potentials. J. Differ. Equ. 249, 2964–2985 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Yuanhong Wei

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Minbo Yang

Authors
  1. Yuanhong Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minbo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanhong Wei.

Additional information

Minbo Yang is supported by ZJNSF (Y7080008), ZJIP (T200905) and NSFC (11101374, 10971194).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wei, Y., Yang, M. Existence of solutions for a system of diffusion equations with spectrum point zero. Z. Angew. Math. Phys. 65, 325–337 (2014). https://doi.org/10.1007/s00033-013-0334-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-013-0334-0

Mathematics Subject Classification (2010)

Keywords

Search

Navigation