Advertisement

A multiplicity result for Chern–Simons–Schrödinger equation with a general nonlinearity

  • Patricia L. Cunha
  • Pietro d’Avenia
  • Alessio Pomponio
  • Gaetano Siciliano
Article

Abstract

In this paper we give a multiplicity result for the following Chern–Simons–Schrödinger equation
$$-\Delta u + 2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds + q u \frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad {\rm in} \mathbb{R}^2,$$
where \({h_u(s) = \int_0^s \tau u^2(\tau)\, d\tau}\), under very general assumptions on the nonlinearity g. In particular, for every \({n \in \mathbb{N}}\), we prove the existence of (at least) n distinct solutions, for every \({q \in (0, q_{n})}\), for a suitable q n .

Keywords

Chern–Simons gauge field Schrödinger equation Variational methods Radial solutions General nonlinearities 

Mathematics Subject Classification

35J20 35Q55 81T10 

References

  1. 1.
    Azzollini A., d’Avenia P., Pomponio A.: Multiple critical points for a class of nonlinear functionals. Ann. Mat. Pura Appl. (4) 190, 507–523 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Berestycki, H., Gallouët, T., Kavian, O.: Équations de Champs scalaires euclidiens non linéaires dans le plan. C R. Acad. Sci. Paris Sér. I Math. 297, 307–310 (1983) and Publications du Laboratoire d’Analyse Numérique, Université de Paris VI (1984)Google Scholar
  3. 3.
    Berestycki H., Lions P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Berestycki H., Lions P.-L.: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82, 347–375 (1983)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Berti M., Bolle P.: Periodic solutions of nonlinear wave equations with general nonlinearities. Commun. Math. Phys. 243, 315–328 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Byeon J., Huh H., Seok J.: Standing waves of nonlinear Schrödinger equations with the gauge field. J. Funct. Anal. 263, 1575–1608 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Deser S., Jackiw R., Templeton S.: Topologically massive gauge theories. Ann. Phys. 140, 372–411 (1982)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Felsager B.: Geometry, Particles, and Fields. Springer, New York (1998)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hagen C.: A new gauge theory without an elementary photon. Ann. Phys. 157, 342–359 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hirata J., Hikoma N., Tanaka K.: Nonlinear scalar field equations in \({\mathbb{R}^N}\): mountain pass and symmetric mountain pass approaches. Topol. Methods Nonlinear Anal. 35, 253–276 (2010)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Huh H.: Standing waves of the Schrödinger equation coupled with the Chern–Simons gauge field. J. Math. Phys. 53, 063702–063708 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jackiw R., Pi S.-Y.: Soliton solutions to the gauged nonlinear Schrödinger equation on the plane. Phys. Rev. Lett. 64, 2969–2972 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Jackiw R., Pi S.-Y.: Classical and quantal nonrelativistic Chern–Simons theory. Phys. Rev. D (3) 42, 3500–3513 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jeanjean L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28, 1633–1659 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jeanjean L., Le Coz S.: An existence and stability result for standing waves of nonlinear Schrödinger equations. Adv. Differ. Equ. 11, 813–840 (2006)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Jeong W., Seok J.: On perturbation of a functional with the mountain pass geometry: applications to the nonlinear Schrödinger–Poisson equations and the nonlinear Klein–Gordon–Maxwell equations. Calc. Var. Partial Differ. Equ. 49, 649–668 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Lerda A.: Anyons: Quantum Mechanics of Particles with Fractional Statistics. Springer, Berlin, Hidelberg (1992)zbMATHGoogle Scholar
  18. 18.
    Naber G.L.: Topology, Geometry, and Gauge Fields: Foundations–Interactions. Springer, New York (2011)Google Scholar
  19. 19.
    Pomponio, A., Ruiz, D.: A Variational Analysis of a Gauged Nonlinear Schrödinger Equation. J. Eur. Math. Soc. (to appear)Google Scholar
  20. 20.
    Pomponio, A., Ruiz, D.: Boundary concentration of a Gauged nonlinear Schrödinger equation on large balls. Calc. Var. Partial Differ. Equ. 53, 289– 316 (2015)Google Scholar
  21. 21.
    Schonfeld J.F.: A mass term for three-dimensional gauge fields. Nucl. Phys. B 185, 157–171 (1981)CrossRefGoogle Scholar
  22. 22.
    Tarantello G.: Self-dual Gauge field Vortices: An Analytical Approach. Birkäuser, Boston (2007)Google Scholar
  23. 23.
    Wan Y., Tan J.: Standing waves for the Chern–Simons–Schrödinger systems without (AR) condition. J. Math. Anal. Appl. 415, 422–434 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Wilczek F.: Fractional Statistics and Anyon Superconductivity. World Scientific, Teaneck (1990)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Patricia L. Cunha
    • 1
  • Pietro d’Avenia
    • 2
  • Alessio Pomponio
    • 2
  • Gaetano Siciliano
    • 3
  1. 1.Departamento de Informática e Métodos QuantitativosFundação Getulio VargasSão PauloBrazil
  2. 2.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly
  3. 3.Departamento de MatemáticaUniversidade de São PauloSão PauloBrazil

Personalised recommendations