Skip to main content
SpringerLink
Log in

Multiplicity of Solutions for a Nonlinear Klein-Gordon-Maxwell System

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper we study the nonlinear Klein-Gordon-Maxwell system

$$\left \{\begin{array}{l@{\quad}l} -\Delta u+V(x)u-(2\omega+\phi)\phi u=f(x,u),&x\in{\mathbb{R}}^3,\\ \Delta \phi=(\omega+\phi)u^2,&x\in{\mathbb{R}}^3. \end{array} \right . $$

By means of a variant fountain theorem and the symmetric mountain pass theorem, we obtain the existence of infinitely many large energy 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. Azzollini, A., Pisani, L., Pomponio, A.: Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system. Proc. R. Soc. Edinb., Sect. A, Math. 141, 449–463 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. Topol. Methods Nonlinear Anal. 35, 33–42 (2010)

    MATH  MathSciNet  Google Scholar 

  3. Azzollini, A., d’Avenia, P., Pomponio, A.: On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, 779–791 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for superlinear elliptic problems on \({\mathbb{R}}^{N}\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bartsch, T., Wang, Z.-Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51, 366–384 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Benci, V., Fortunato, D.: The nonlinear Klein-Gordon equation coupled with the Maxwell equations. Nonlinear Anal. 47, 6065–6072 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Benci, V., Fortunato, D.: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carrião, P., Cunha, P., Miyagaki, O.: Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents. Commun. Pure Appl. Anal. 10, 709–718 (2011)

    MATH  MathSciNet  Google Scholar 

  11. Carrião, P., Cunha, P., Miyagaki, O.: Positive and ground state solutions for the critical Klein-Gordon-Maxwell system with potentials. Preprint arXiv:1005.4088 [math.AP]

  12. Cassani, D.: Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations. Nonlinear Anal. 58, 733–747 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. D’Aprile, T., Mugnai, D.: Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)

    MATH  MathSciNet  Google Scholar 

  14. D’Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. R. Soc. Edinb., Sect. A, Math. 134, 1–14 (2004)

    Article  MathSciNet  Google Scholar 

  15. Georgiev, V., Visciglia, N.: Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential. J. Math. Pures Appl. 84, 957–983 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. American Mathematical Society, Providence (1986)

    Google Scholar 

  19. Willem, M.: Minimax Theorems. Birkhaüser, Basel (1996)

    Book  MATH  Google Scholar 

  20. Zou, W.: Variant fountian theorem and their applications. Manuscr. Math. 104, 343–358 (2001)

    Article  MATH  Google Scholar 

  21. Zou, W., Schechter, M.: Critical Point Theory and Its Applications. Springer, New York (2006)

    MATH  Google Scholar 

Download references

Acknowledgements

The author is very grateful to the anonymous referee for his/her careful reading the manuscript and valuable comments. This work was supported by NSFC Grants (11271386).

Author information

Authors and Affiliations

  1. College of Sciences, Minzu University of China, Beijing, 100081, P.R. China

    Xiaoming He

Authors
  1. Xiaoming He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoming He.

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, X. Multiplicity of Solutions for a Nonlinear Klein-Gordon-Maxwell System. Acta Appl Math 130, 237–250 (2014). https://doi.org/10.1007/s10440-013-9845-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-013-9845-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation