Skip to main content
SpringerLink
Log in

Bound and Ground States for a Concave–Convex Generalized Choquard Equation

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

For a class of generalized Choquard equations with perturbation terms, we establish the existence of bound and ground states by virtue of the method of Nehari manifold.

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., Yang, M.B.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257, 4133–4164 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Bartsch, T., Wang, Z.-Q., Willem, M.: The Dirichlet problem for superlinear elliptic equations. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 2. Elsevier, Amsterdam (2005)

    Google Scholar 

  4. Brown, K.J., Wu, T.-F.: A fibering map approach to a semilinear elliptic boundary value problem. Electron. J. Differ. Equ., No. 69, 9 pp. (2007)

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

    Article  MATH  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. Edinb. A 140, 973–1009 (2010)

    Article  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Goyal, S., Sreenadh, K.: Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci. Math. Sci. 125, 545–558 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/77)

  11. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics. Am. Math. Soc., Providence (2001)

    Book  MATH  Google Scholar 

  12. Lions, P.L.: The Choquard equation and related equations. Nonlinear Anal. 4, 1063–1073 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Moroz, I.M., Tod, P.: An analytical approach to the Schrödinger–Newton equations. Nonlinearity 12, 201–216 (1999)

    Article  MATH  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  MATH  Google Scholar 

  16. Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equation. Calc. Var. Partial Differ. Equ. 52, 199–235 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  17. Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, 281–304 (1992)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Yang, M., Ding, Y.: Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Commun. Pure Appl. Anal. 12, 771–783 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhang, Z.J., Tassilo, K., Hu, A.L., Xia, H.Q.: Existence of a nontrivial solution for Choquard’s equation. Acta Math. Sci. Ser. B Engl. Ed. 26, 460–468 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their careful reading and helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, China

    Hui Zhang

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Junxiang Xu & Fubao Zhang

Authors
  1. Hui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fubao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Zhang.

Additional information

The work was supported by the National Natural Science Foundation of China (No. 11601204), the Special Funds of the National Natural Science Foundation of China (No. 11526103), the Natural Science Foundation of Jiangsu Province (No. BK20140106) and the Research Foundation of Jinling Institute of Technology (No. jit-b-201403).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Xu, J. & Zhang, F. Bound and Ground States for a Concave–Convex Generalized Choquard Equation. Acta Appl Math 147, 81–93 (2017). https://doi.org/10.1007/s10440-016-0069-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-016-0069-y

Keywords

Mathematics Subject Classification

Search

Navigation