Monatshefte für Mathematik

, Volume 182, Issue 2, pp 335–358 | Cite as

Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems

  • Dengfeng Lü


We consider the following singularly perturbed nonlocal elliptic problem
$$\begin{aligned} -\left( \varepsilon ^{2}a+\varepsilon b\displaystyle \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u+V(x)u=\displaystyle \varepsilon ^{\alpha -3}(W_{\alpha }(x)*|u|^{p})|u|^{p-2}u, \quad x\in \mathbb {R}^{3}, \end{aligned}$$
where \(\varepsilon >0\) is a parameter, \(a>0,b\ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2, 6-\alpha )\), \(W_{\alpha }(x)\) is a convolution kernel and V(x) is an external potential satisfying some conditions. By using variational methods, we establish the existence and concentration of positive ground state solutions for the above equation.


Kirchhoff-type equation Hartree-type nonlinearity  Variational method Ground state solution Concentration phenomena 

Mathematics Subject Classification

35J60 35B25 35B40 



The author is grateful to Professor Shuangjie Peng for his helpful suggestions and careful guidance.


  1. 1.
    Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alves, C.O., Correa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alves, C.O., Yang, M.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257, 4133–4164 (2014)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cingolani, S., Secchi, S., Squassina, M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinb. Sect. A 140, 973–1009 (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Clapp, M., Salazar, D.: Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl. 407, 1–15 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    He, Y., Li, G., Peng, S.: Concentrating bound states for Kirchhoff type problem in \(\mathbb{R}^{3}\) involving critical Sobolev exponents. Adv. Nonlinear Stud. 14, 441–468 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^{3}\). J. Differ. Equ. 252, 1813–1834 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)zbMATHGoogle Scholar
  10. 10.
    Lieb, E.H., Loss, M.: Analysis. In: Graduate Studies in Mathematics, vol. 14, 2nd edn. AMS, Providence (2001)Google Scholar
  11. 11.
    Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lieb, E.H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lions, J.L.: On some questions in boundary value problems of mathematical physics. North-Holl. Math. Stud. 30, 284–346 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lions, P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, W., He, X.: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, 473–487 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lü, D.: A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 99, 35–48 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moroz, V., Van Schaftingen, J.: Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)CrossRefzbMATHGoogle Scholar
  19. 19.
    Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equation. Calc. Var. Partial Differ. Equ. 52, 199–235 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581–600 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)CrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, J., Tian, L., Xu, J., Zhang, F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser, Boston (1996)Google Scholar
  24. 24.
    Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(R^{N}\). Nonlinear Anal. Real World Appl. 12, 1278–1287 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral China Normal UniversityWuhanPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHubei Engineering UniversityXiaoganPeople’s Republic of China

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