Skip to main content
SpringerLink
Log in

On the nonlinear Schrödinger–Poisson systems with sign-changing potential

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we study a nonlinear Schrödinger–Poisson system

$$\left\{ \begin{array}{ll} -\Delta u+V_{\lambda }( x) u+\mu K( x) \phi u=f (x, u) & \text{in}\;\mathbb{R}^{3}, \\ -\Delta \phi =K ( x ) u^{2} & \text{in}\;\mathbb{R}^{3},\end{array}\right.$$

where \({\mu > 0}\) is a parameter, \({V_{\lambda }}\) is allowed to be sign-changing and f is an indefinite function. We require that \({V_{\lambda }:=\lambda V^{+}-V^{-}}\) with V + having a bounded potential well Ω whose depth is controlled by λ and \({V^{-} \geq 0}\) for all \({x\in \mathbb{R} ^{3}}\). Under some suitable assumptions on K and f, the existence and the nonexistence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is explored as well.

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. Alves C.O., Carrião P.C., Medeiros E.S.: Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions. Abstr. Appl. Anal. 3, 251–268 (2004)

    Article  Google Scholar 

  2. Ambrosetti A., Ruiz D.: Multiple bound states for the Schrödinger–Poisson problem. Commun. Contemp. Math. 10(3), 39–404 (2008)

    Article  MathSciNet  Google Scholar 

  3. Azzollini A.: Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity. J. Differ. Equ. 249, 1746–1763 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartsch T., Pankov A., Wang Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 549–569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bartsch T., Tang Z.: Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete Contin. Dyn. Syst. 33, 7–26 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Bartsch T., Wang Z.-Q.: Existence and multiplicity results for superlinear elliptic problems on \({\mathbb{R}^{N}}\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Brezis H., Lieb E.H.: A relation between pointwise convergence of functions and convergence functionals. Proc. Am. Math. Soc. 8, 486–490 (1983)

    Article  MathSciNet  Google Scholar 

  10. Chen C.Y., Kuo Y.C., Wu T.F.: Existence and multiplicity of positive solutions for the nonlinear Schrödinger–Poisson equations. Proc. R. Soc. Edinb. Sect. A 143, 745–764 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Ding Y.H., Szulkin A.: Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. Partial Differ. Equ. 29, 397–419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ekeland I.: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  15. de Figueiredo D.G.: Positive Solutions of Semilinear Elliptic Problems, Lecture in Math. vol. 957. Springer, Berlin (1982)

    Google Scholar 

  16. Ianni I.: Sign-changing radial solutions for the Schrödinger–Poisson–Slater problem. Topol. Methods Nonlinear Anal. 41, 365–385 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Ianni I., Ruiz D.: Ground and bound states for a static Schrödinger–Poisson–Slater problem. Commun. Contemp. Math. 14, 1250003 (2012)

    Article  MathSciNet  Google Scholar 

  18. Ianni I., Vaira G.: On concentration of positive bound states for the Schrödinger–Poisson problem with potentials. Adv. Nonlinear Stud. 8, 573–595 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Jiang Y., Zhou H.: Schrödinger–Poisson system with steep potential well. J. Differ. Equ. 251, 582–608 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. H. Poincare Anal. NonLineaire 1, 109–145 (1984)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Ruiz D.: On the Schrödinger–Poisson–Slater System: behavior of minimizers, radial and nonradial cases. Arch. Ratl. Mech. Anal. 198, 349–368 (2010)

    Article  MATH  Google Scholar 

  25. Sato Y., Tanaka K.: Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells. Trans. Am. Math. Soc. 361, 6205–6253 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sun J.: Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 390, 514–522 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun J., Chen H., Nieto J.J.: On ground state solutions for some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 252, 3365–3380 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sun J., Wu T.F.: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 256, 1771–1792 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ye, Y., Tang, C.: Existence and multiplicity of solutions for Schrödinger–Poisson equations with sign-changing potential. Calc. Var. Partial Differ. Equ. doi:10.1007/s00526-014-0753-6

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang Z., Zhou H.: Positive solutions for nonlinear Schrödinger equations with deepening potential well. J. Eur. Math. Soc. 11, 545–573 (2009)

    Article  MATH  Google Scholar 

  32. Zhao L., Liu H., Zhao F.: Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J. Differ. Equ. 255, 1–23 (2013)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Shandong University of Technology, Zibo, 255049, People’s Republic of China

    Juntao Sun

  2. Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan

    Tsung-fang Wu

Authors
  1. Juntao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsung-fang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tsung-fang Wu.

Additional information

J. Sun was supported by the National Natural Science Foundation of China (Grant Nos. 11201270 and 11271372), Shandong Natural Science Foundation (Grant No. ZR2012AQ010), Young Teacher Support Program of Shandong University of Technology, and China Postdoctoral Science Foundation (Grant No. 2014M551494). T.F. Wu was supported in part by the National Science Council and the National Center for Theoretical Sciences, Taiwan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, J., Wu, Tf. On the nonlinear Schrödinger–Poisson systems with sign-changing potential. Z. Angew. Math. Phys. 66, 1649–1669 (2015). https://doi.org/10.1007/s00033-015-0494-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-015-0494-1

Mathematical Subject Classification

Keywords

Search

Navigation