Skip to main content
SpringerLink
Log in

Solitons in Schrödinger-Maxwell equations

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper we study the nonlinear Schrödinger–Maxwell equations (NSM). We are interested in analyzing the existence of solitons, namely the existence of finite energy solutions which exhibit stability properties. This paper is divided into two parts. In the first part, we give an abstract definition of solitons and we develop an abstract existence theory. In the second, we apply this theory to NSM.

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. Ambrosetti A.: On Schrödinger-Poisson systems. Milan J. Math. 76, 257–274 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosetti A., Ruiz D.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10, 391–404 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bellazzini J., Benci V., Bonanno C., Micheletti A.M.: Solitons for the nonlinear Klein-Gordon equation. Adv. Nonlinear Stud. 10, 481–499 (2010)

    MATH  MathSciNet  Google Scholar 

  4. Bellazzini J., Benci V., Bonanno C., Sinibaldi E.: Hylomorphic solitons in the nonlinear Klein-Gordon equation. Dyn. Partial Differ. Equ. 6, 311–334 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bellazzini J., Benci V., Ghimenti M., Micheletti A. M.: On the existence of the fundamental eigenvalue of an elliptic problem in \({\mathbb{R}^{N}}\). Adv. Nonlinear Stud. 7, 439–458 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Bellazzini J., Siciliano G.: Stable standing waves for a class of nonlinear Schrödinger-Poisson equations. Z. Angew. Math. Phys. 62, 267–280 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Benci V.: Hylomorphic solitons. Milan J. Math. 77, 271–332 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  9. Benci V., Fortunato D.: Solitary waves in the nonlinear wave equation and in gauge theories. J. Fixed Point Theory Appl. 1, 61– (2007)

  10. Benci V., Fortunato D.: Hylomorphic solitons on lattices. Discrete Contin. Dyn. Syst. 28, 875–897 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. V. Benci and D. Fortunato, On the existence of stable charged Q-balls. J. Math. Phys. 52 (2011), doi:10.1063/1.3629848.

  12. V. Benci and D. Fortunato, Hamiltonian formulation of the Klein-Gordon-Maxwell equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011), 113-134.

  13. V. Benci and D. Fortunato, A minimization method and applications to the study of solitons. Nonlinear Anal. 75 (2012), 4398-4421.

  14. V. Benci and D. Fortunato, Hylomorphic solitons and charged Q-balls: Existence and stability. Chaos Solitons Fractals 58 (2014), 1-15.

  15. V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations. Springer Monographs in Mathematics, Springer, New York, 2014.

  16. A. M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations. Mediterr. J. Math. 3 (2006), 483-493.

  17. T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982), 549-561.

  18. Cerami G., Vaira G.: Positive solutions for some non-autonomous Schrödinger-Poisson systems. J. Differential Equations 248, 521–543 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun. Appl. Anal. 7 (2003), 417-423.

  20. G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations. Electron. J. Differential Equations 94 (2004), 1-31.

  21. T. D’Aprile, Semiclassical states for the nonlinear Schrödinger equations with the electromagnetic field. NoDEA Nonlinear Differential Equations Appl. 13 (2007), 655-681.

  22. T. D’Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893-906.

  23. T. D’Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation. SIAM J. Math. Anal. 37 (2005), 321-342.

  24. T. D’Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem. Calc. Var. Partial Differential Equations 25 (2006), 105-137.

  25. P. d’Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear Stud. 2 (2002) 177-192.

  26. P. d’Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system. Nonlinear Anal. 74 (2011), 5705-5721.

  27. I. M. Gelfand and S. V. Fomin, Calculus of Variations. Prentice-Hall, Englewood Cliffs, NJ, 1963.

  28. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), 160-197.

  29. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94 (1990), 308-348.

  30. Y. Guo, K. Nakamitsu and W. Strauss, Global finite energy solutions of the Maxwell-Schrödinger system. Comm. Math. Phys. 170 (1995), 181-196.

  31. I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potential. Adv. Nonlinear Stud. 8 (2008), 573-595.

  32. Kikuchi H.: Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7, 403–437 (2007)

    MATH  MathSciNet  Google Scholar 

  33. H. Kikuchi, On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal. 67 (2007), 1445-1456.

  34. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109-145.

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

  36. M. Nakamitsu and M. Tsutsumi, The Cauchy problem for the coupled Maxwell-Schrödinger equations. J. Math. Phys. 27 (1986), 211-216.

  37. M. Nakamura and T. Wada, Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations. Comm. Math. Phys. 276 (2007), 315-339.

  38. Nirenberg L.: On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa (3(13), 115–162 (1959)

    MathSciNet  Google Scholar 

  39. L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system. Topol. Methods Nonlinear Anal. 29 (2007), 251-264.

  40. V. Rubakov, Classical Theory of Gauge Fields. Princeton University Press, Princeton, NJ, 2002.

  41. D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere. Math. Models Methods Appl. Sci. 15 (2005), 141-164.

  42. D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237 (2006), 655-674.

  43. D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential. Rev. Mat. Iberoam. 27 (2011), 253-271.

  44. A. Salvatore, Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in \({{\mathbb{R}^{3}}}\). Adv. Nonlinear Stud. 6 (2006), 157-169.

  45. Sanchez O., Soler J.: Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Stat. Phys. 114, 179–204 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  46. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Springer, New York, 1999.

  47. M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), 51-67.

  48. Y. Yang, Solitons in Field Theory and Nonlinear Analysis. Springer, New York, 2001.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, Pisa, Italy

    Vieri Benci

  2. Dipartimento di Matematica, Università degli Studi di Bari “Aldo Moro” and INFN sezione di Bari, Via Orabona 4, Bari, Italy

    Donato Fortunato

Authors
  1. Vieri Benci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donato Fortunato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vieri Benci.

Additional information

To Haïm Brezis with friendship and admiration

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Benci, V., Fortunato, D. Solitons in Schrödinger-Maxwell equations. J. Fixed Point Theory Appl. 15, 101–132 (2014). https://doi.org/10.1007/s11784-014-0184-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-014-0184-1

Mathematics Subject Classification

Keywords

Search

Navigation