Skip to main content
SpringerLink
Log in

Infinitely Many Solutions for the Klein–Gordon Equation with Sublinear Nonlinearity Coupled with Born–Infeld Theory

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

This paper is concerned with the following Klein–Gordon equation with sublinear nonlinearity coupled with Born–Infeld theory:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+ V(x)u-(2\omega +\phi )\phi u=f(x,u), &{}\quad x\in {\mathbb {R}}^{3},\\ \Delta \phi +\beta \Delta _{4}\phi =4\pi (\omega +\phi )u^{2},&{}\quad x\in {\mathbb {R}}^{3}. \end{array} \right. \end{aligned}$$

Under some appropriate assumptions on V(x) and f(x, u), we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are improved and extended.

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.

Similar content being viewed by others

References

  1. Ambrosetti, A., Malchiodi, A.: Nonlinear analysis and semilinear elliptic problems. in: Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge (2007)

  2. Bartsch, T., Wang, Z.: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}}^{N}\). Commun. Part. Diff. Equat. 20, 1725–1741 (1995)

    Article  Google Scholar 

  3. Benci, V., Fortunato, D., Masiello, A., Pisani, L.: Solitons and the electromagnetic field. Math. Z. 232, 73–102 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Born, M., Infeld, L.: Foundations of the new field theory. Proc. R. Soc. Lond. A. 144, 425–451 (1934)

    Article  Google Scholar 

  6. 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  MathSciNet  Google Scholar 

  7. Carriao, 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)

    MathSciNet  MATH  Google Scholar 

  8. Che, G., Chen, H.: Infinitely many solutions for a class of modified nonlinear fourth-order elliptic equations on \({\mathbb{R}}^{N}\). Bull. Korean Math. Soc. 54, 895–909 (2017)

    Article  MathSciNet  Google Scholar 

  9. Che, G., Chen, H.: Existence and multiplicity of nontrivial solutions for Klein-Gordon-Maxwell system with a parameter. J. Korean Math. Soc. 54, 1015–1030 (2017)

    Article  MathSciNet  Google Scholar 

  10. Che, G., Chen, H.: Infinitely many solutions of systems of Kirchhoff-type equations with general potentials. Rocky Mt. J. Math. 48, 2187–2209 (2018)

    Article  MathSciNet  Google Scholar 

  11. Che, G., Chen, H., Wu, T.F.: Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling. J. Math. Phys. 60, 081511 (2019)

    Article  MathSciNet  Google Scholar 

  12. Che, G., Shi, H., Wang, Z.: Existence and concentration of positive ground states for a 1-Laplacian problem in \({\mathbb{R}}^{N}\). Appl. Math. Lett. 100, 106045 (2020)

  13. Chen, S., Li, L.: Multiple solutions for the nonhomogeneous Klein-Gordon equation coupled with Born-Infeld theory on \({\mathbb{R}}^{3}\). J. Math. Anal. Appl. 400, 517–524 (2013)

    Article  MathSciNet  Google Scholar 

  14. Chen, S., Song, S.: The existence of multiple solutions for the Klein-Gordon equation with concave and convex nonlinearities coupled with Born-Infeld theory on \({\mathbb{R}}^{3}\). Nonlinear Anal. 38, 78–95 (2017)

    Article  MathSciNet  Google Scholar 

  15. Daprile, T., Mugnai, D.: Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)

    MathSciNet  MATH  Google Scholar 

  16. d’Avenia, P., Pisani, L.: Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations. Electron. J. Differ. Equ. 26, 1–13 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Felsager, B.: Geometry, Particles and Fields, Odense University Press, Odense (1981) Edited with the assistance of Carsten Claussen

  18. Figueiredo, G. M., Siciliano, G.: Existence and asymptotic behaviour of solutions for a quasi-linear Schrödinger-Poisson system under a critical nonlinearity. arXiv:1707.05353

  19. Liu, H., Chen, H.: Multiple solutions for superlinear Schrödinger-Poisson system with sign-changing potential and nonlinearity. Comput. Math. Appl. 68, 1982–1990 (2014)

    Article  MathSciNet  Google Scholar 

  20. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74. Springer, New York (1989)

    Book  Google Scholar 

  21. Mugnai, D.: Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves. Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 460, 1519–1527 (2004)

    Article  MathSciNet  Google Scholar 

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

  23. Shi, H., Chen, H.: Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth. Comput. Math. Appl. 71, 349–358 (2016)

    Article  MathSciNet  Google Scholar 

  24. Shi, H., Chen, H.: Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations. J. Math. Anal. Appl. 452, 578–594 (2017)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  26. Tang, X., Lin, H.: Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials. Nonlinear Anal. 74, 6314–6325 (2011)

    Article  MathSciNet  Google Scholar 

  27. Teng, K.: Existence and multiplicity of solutions for the nonlinear Klein-Gordon equation coupled with Born-Infeld theory on bounded domain. J. Difference Equ. Appl. 4, 445–457 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Teng, K., Zhang, K.: Existence of solitary wave solutions for the nonlinear Klein-Gordon equation coupled with Born-Infeld theory with critical Sobolev exponent. Nonlinear Anal. 74, 4241–4251 (2011)

    Article  MathSciNet  Google Scholar 

  29. Willem, M.: Minimax Theorems. Birkhäuser Boston Inc, Boston (1996)

    Book  Google Scholar 

  30. Xu, L., Chen, H.: Nontrivial solutions for Kirchhoff-type problems with a parameter. J. Math. Anal. Appl. 433, 455–472 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, People’s Republic of China

    Guofeng Che

  2. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Haibo Chen

Authors
  1. Guofeng Che
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haibo Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guofeng Che.

Additional information

Communicated by Asadollah Aghajani.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by National Natural Science Foundation of China 11671403.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Che, G., Chen, H. Infinitely Many Solutions for the Klein–Gordon Equation with Sublinear Nonlinearity Coupled with Born–Infeld Theory . Bull. Iran. Math. Soc. 46, 1083–1100 (2020). https://doi.org/10.1007/s41980-019-00314-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-019-00314-3

Keywords

Mathematics Subject Classification

Search

Navigation