Skip to main content
SpringerLink
Log in

Existence and Concentration of Solutions for a Nonlinear Choquard Equation

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the nonlinear Choquard equation

$$-\Delta u+(1 + \mu g(x))u = (K_{\alpha}(x) * |u|^{p} ) |u|^{p-2}u, \ \ \ x \in \mathbb{R}^{N},$$

where \({N \geq 3}\) , \({\alpha \in (0, N)}\) , \({p \in (\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2})}\) , \({\mu > 0}\) is a parameter, \({K_{\alpha}(x)}\) is the Riesz potential and g(x) is a nonnegative continuous potential. Under some assumptions on g(x), we obtain the existence of ground state solutions and concentration results by using the critical point theory.

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. Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z 248, 423–443 (2004)

    Article  MathSciNet  Google Scholar 

  2. Brézis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  Google Scholar 

  3. Bartsch T., Wang Z.Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51, 366–384 (2000)

    Article  MathSciNet  Google Scholar 

  4. Cingolani S., Clapp M., Secchi S.: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63, 233–248 (2012)

    Article  MathSciNet  Google Scholar 

  5. Cingolani S., Clapp M., Secchi S.: Intertwining semiclassical solutions to a Schrödinger–Newton system. Discret. Contin. Dyn. Syst. Ser. S 6, 891–908 (2013)

    MathSciNet  Google Scholar 

  6. Cingolani S., Secchi S., Squassina M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinburgh Sect. A 140, 973–1009 (2010)

    Article  MathSciNet  Google Scholar 

  7. Clapp M., Salazar D.: Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl. 407, 1–15 (2013)

    Article  MathSciNet  Google Scholar 

  8. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics 14, AMS, USA (2001)

  9. Lieb E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)

    MathSciNet  Google Scholar 

  10. Lieb E.H., Simon B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys 53, 185–194 (1977)

    Article  MathSciNet  Google Scholar 

  11. Lions P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  13. Lü D.: A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 99, 35–48 (2014)

    Article  MathSciNet  Google Scholar 

  14. Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)

    Article  MathSciNet  Google Scholar 

  15. Moroz V., Van Schaftingen J.: Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal 265, 153–184 (2013)

    Article  MathSciNet  Google Scholar 

  16. Nolasco M.: Breathing modes for the Schrödinger–Poisson system with a multiple-well external potential. Commun. Pure Appl. Anal. 9, 1411–1419 (2010)

    Article  MathSciNet  Google Scholar 

  17. Penrose R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581–600 (1996)

    Article  MathSciNet  Google Scholar 

  18. Wei J., Winter M.: Strongly interacting bumps for the Schrödinger–Newton equations. J. Math. Phys. 50, 012905 (2009)

    Article  MathSciNet  Google Scholar 

  19. Willem, M.: Minimax theorems. In: Progress in nonlinear differential equations and their applications, vol. 24. Birkhäuser, Boston (1996)

  20. Yang M., Wei Y.: Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities. J. Math. Anal. Appl. 403, 680–694 (2013)

    Article  MathSciNet  Google Scholar 

  21. Zhang Z., Tassilo K., Hu A., Xia H.: Existence of a nontrivial solution for Choquard’s equation. Acta Math. Sci. 26B(3), 460–468 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000, Hubei, People’s Republic of China

    Dengfeng Lü

Authors
  1. Dengfeng Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dengfeng Lü.

Additional information

This work was supported by the science and technology research project of Hubei Provincial Department of Education (D20142702).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lü, D. Existence and Concentration of Solutions for a Nonlinear Choquard Equation. Mediterr. J. Math. 12, 839–850 (2015). https://doi.org/10.1007/s00009-014-0428-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-014-0428-8

Mathematics Subject Classification (2010)

Keywords

Search

Navigation