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Stable standing waves for a class of nonlinear Schrödinger-Poisson equations

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Abstract

We prove the existence of orbitally stable standing waves with prescribed L 2-norm for the following Schrödinger-Poisson type equation

$$i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 \quad \rm{in} \quad \mathbb R^{3},$$

when \({p\in \left\{ \frac{8}{3}\right\}\cup (3,\frac{10}{3})}\). In the case \({3 < p < \frac{10}{3}}\), we prove the existence and stability only for sufficiently large L 2-norm. In case \({p=\frac{8}{3}}\), our approach recovers the result of Sanchez and Soler (J Stat Phys 114:179–204, 2004) for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In the final section, a further application to the Schrödinger equation involving the biharmonic operator is given.

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References

  1. Azzollini A., Pomponio A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345(1), 90–108 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellazzini, J., Bonanno, C.: Nonlinear Schrödinger equations with strongly singular potentials. Proc. Roy. Soc. Edimburgh sec A (in press), arXiv:0903.3301

  3. Bellazzini, J., Visciglia, N.: On the orbital stability for a class of nonautonmous NLS. Indiana Univ. Math. J. 59(2), (2010) online preprint

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

    MATH  MathSciNet  Google Scholar 

  5. Brezis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88(3), 486–490 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cazenave, T.: Semilinear Schrödinger equation, Courant. In: Lecture Notes in Mathematics, vol. 10, New York: New York University, Courant Institute of Mathematical Sciences

  7. Cazenave T., Lions P.L.: Orbital stability of standing waves for some non linear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen G., Zhang J., Wei Y.: Stability of standing waves for nonlinear defocusing fourth-order dispersive Schrödinger equation with unbounded potentials. Math. Nachr. 281(4), 517–524 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. D’Aprile T., Mugnai D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Royal Soc. Edinb. 134 A, 893–906 (2004)

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  12. Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ianni I., Le Coz S.: Orbital stability of standing waves of a semiclassical nonlinear Schrö dinger-Poisson equation. Adv. Differ. Equ. 14(7–8), 717–748 (2009)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Kikuchi H.: Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7(3), 403–437 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Lions, P.L.: The concentration-compactness principle in the calculus of variation. The locally compact case, part I and II. Ann. Inst. H. Poincar Anal. Non Linaire 1, 109–145 and 223–283 (1984)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. 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 

  20. Wang Z., Zhou H.S.: Positive solution for a nonlinear stationary Schrödinger-Poisson system in \({\mathbb{R}^{3}}\). Discrete Contin. Dyn. Syst. 18, 809–816 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zhao L., Zhao F.: On the existence of solutions for the Schrödinger-Poisson equations. J. Math. Anal. Appl 346, 155–169 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata “U. Dini”, University of Pisa, via Buonarroti 1/c, 56127, Pisa, Italy

    Jacopo Bellazzini

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, Brasil

    Gaetano Siciliano

Authors
  1. Jacopo Bellazzini
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  2. Gaetano Siciliano
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Correspondence to Jacopo Bellazzini.

Additional information

The authors are partially supported by M.I.U.R project PRIN2007 “Variational and topological methods in the study of nonlinear phenomena”.

The second author is also supported by J. Andalucía (FQM 116) and by FAPESP Grant 2010/00068-6.

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Bellazzini, J., Siciliano, G. Stable standing waves for a class of nonlinear Schrödinger-Poisson equations. Z. Angew. Math. Phys. 62, 267–280 (2011). https://doi.org/10.1007/s00033-010-0092-1

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  • DOI: https://doi.org/10.1007/s00033-010-0092-1

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