Advertisement

Multiple normalized solutions of Chern–Simons–Schrödinger system

  • Jianjun YuanEmail author
Article

Abstract

In this paper, we consider the following equation
$$-\Delta u + \omega u + \left(\int_{|x|}^{\infty}\frac{h(s)}{s}u^2(s)ds\right) u + \frac{h^2(|x|)}{|x|^2}u - \lambda|u|^{p - 2}u = 0\quad \mbox{in}\quad \mathbb{R}^2,$$
for p >  2 and \({\lambda > 0}\) , which appeared in Byeon et al. (J Funct Anal 263(6):1575–1608, 2012) to find the standing wave solutions of the Chern–Simons–Schrödinger system. By using the minimax theorem, we get the multiplicity results for the L 2-normalized solutions to the equation, and thus there are multiple L 2-normalized solutions of the Chern–Simons–Schrödinger system.

Keywords

Palais–Smale sequence Euler–Lagrange equation Pohozaev identity Standing wave 

Mathematics Subject Classification

35Q55 35A15 35B30 

References

  1. 1.
    Ambrosetti A.: On Schrödinger–Poisson systems. Milan J. Math. 76, 257–274 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosetti A., Ruiz D.: Multiple bound states for the Schrödinger–Poisson problem. Commun. Contemp. Math. 10(3), 391–404 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bartsch T., De Valeriola S.: Normalized solutions of nonlinear Schrödinger equations. Arch. Math. 100, 75–83 (2013)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bergé, L., de Bouard, A., Saut, J.C.: Blowing up time-dependent solutions of the planar Chern–Simons gauged nonlinear Schrödinger equation. Nonlinearity 8(2), 235–253(19) 1995Google Scholar
  5. 5.
    Byeon J., Huh H., Seok J.: Standing waves of nonlinear Schrödinger equations with the gauge field. J. Funct. Anal. 263(6), 1575–1608 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bellazzini J., Jeanjean L., Luo T.-J.: Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations. Proc. Lond. Math. Soc. (3) 107, 303–339 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bellazzini J., Siciliano G.: Stable standing waves for a class of nonlinear Schrödinger–Poisson equations. Zeitschrift für Angewandte Mathematik und Physik. 62(2), 267–280 (2011a)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bellazzini J., Siciliano G.: Scaling properties of functionals and existence of constrained minimizers. J. Funct. Anal. 261(9), 2486–2507 (2011b)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dunne, G.V.: Self-dual Chern-Simons theories. In: Lecture Notes in Physics Monographs, vol. 36. Springer, Berlin (1995)Google Scholar
  10. 10.
    Huh H.: Blow-up solutions of the Chern–Simons–Schrödinger equations. Nonlinearity 22, 967–974 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Huh H.: Standing waves of the Schröinger equation coupled with the Chern–Simons gauge field. J. Math. Phys. 53, 063702 (2012). doi: 10.1063/1.4726192 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huh, H.: Energy solution to the Chern–Simons–Schrödinger equations. In: Crasta, G. (ed.) Abstract and Applied Analysis, vol. 2013, p. 7. Article ID 590653 (2013)Google Scholar
  13. 13.
    Huang, Y., Liu, Z., Wu, Y.: Existence of prescribed L 2-norm solutions for a class of Schrödinger–Poisson equation, vol. 2013, p. 11. Article ID 398164. In: Abstract and Applied Analysis. doi: 10.1155/2013/398164
  14. 14.
    Jeanjean L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. Theory Methods Appl. 28(10), 1633–1659 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jackiw R., Pi S.-Y.: Classical and quantal nonrelativistic Chern–Simons theory. Phys. Rev. D. 42, 3500–3513 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jackiw R., Pi S.-Y.: Self-dual Chern–Simons solitons. Prog. Theor. Phys. Suppl. 107, 1–40 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jeanjean L., Luo T.: Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations. Zeitschrift für angewandte Mathematik und Physik. 64(4), 937–954 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Lions P.-L.: The concentration-compactness principle in the calculus of variation. The locally compact case, part I. Annales de l’Institut Henri Poincare. 1(2), 109–145 (1984a)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Lions P.-L.: The concentration-compactness principle in the calculus of variation. The locally compact case, part II. Annales de l’Institut Henri Poincare. 1(4), 223–283 (1984b)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Luo T.-J.: Multiplicity of normalized solutions for a class of nonlinear Schrödinger–Poisson–Slater equations. J. Math. Anal. Appl. 416(1), 195–204 (2014). doi: 10.1016/j.jmaa.2014.02.038 zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Pomponio, A., Ruiz, D.: A variational analysis of a gauged nonlinear Schrödinger equation. Preprint. arXiv:1306.2051
  22. 22.
    Ruiz D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237(2), 655–674 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Struwe, M: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edn, 34. Springer-Verlag, Berlin (1996)Google Scholar
  24. 24.
    Strauss W.-A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)zbMATHCrossRefGoogle Scholar
  25. 25.
    Willem M.: Minimax Methods. Birkhäuser, Boston (1996)Google Scholar
  26. 26.
    Zhao L., Zhao F.: Positive solutions for Schrödinger–Poisson equations with a critical exponent. Nonlinear Anal. Theory Methods Appl. 70(6), 2150–2164 (2009). doi: 10.1016/j.na.2008.02.116 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.The College of Information and TechnologyNanjing University of Chinese MedicineNanjingChina

Personalised recommendations