Mediterranean Journal of Mathematics

, Volume 12, Issue 3, pp 839–850 | Cite as

Existence and Concentration of Solutions for a Nonlinear Choquard Equation

  • Dengfeng Lü


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.

Mathematics Subject Classification (2010)

35J60 35Q55 35B38 


Nonlinear Choquard equation Critical point theory Ground state solution Concentration behavior 


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  1. 1.
    Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z 248, 423–443 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brézis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)CrossRefGoogle Scholar
  3. 3.
    Bartsch T., Wang Z.Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51, 366–384 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cingolani S., Clapp M., Secchi S.: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63, 233–248 (2012)MathSciNetCrossRefGoogle Scholar
  5. 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)MathSciNetGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clapp M., Salazar D.: Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl. 407, 1–15 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics 14, AMS, USA (2001)Google Scholar
  9. 9.
    Lieb E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)MathSciNetGoogle Scholar
  10. 10.
    Lieb E.H., Simon B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys 53, 185–194 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lions P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)MathSciNetCrossRefGoogle Scholar
  12. 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. 13.
    Lü D.: A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 99, 35–48 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetCrossRefGoogle Scholar
  15. 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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nolasco M.: Breathing modes for the Schrödinger–Poisson system with a multiple-well external potential. Commun. Pure Appl. Anal. 9, 1411–1419 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Penrose R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581–600 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wei J., Winter M.: Strongly interacting bumps for the Schrödinger–Newton equations. J. Math. Phys. 50, 012905 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Willem, M.: Minimax theorems. In: Progress in nonlinear differential equations and their applications, vol. 24. Birkhäuser, Boston (1996)Google Scholar
  20. 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)MathSciNetCrossRefGoogle Scholar
  21. 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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHubei Engineering UniversityXiaoganPeople’s Republic of China

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