Skip to main content
SpringerLink
Log in

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy–Littlewood–Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

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., Figueiredo, G.M., Furtado, M.F.: On the number of solutions of NLS equations with magnetics fields in expanding domains. J. Differ. Equ. 251, 2534–2548 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves, C.O., Figueiredo, G.M., Yang, M.: Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field. Asympt. Anal. 96, 135–159 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, vol. 104. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  4. Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41(3), 253–294 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benci, V., Bonanno, C., Micheletti, A.M.: On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds. J. Funct. Anal. 252, 464–489 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benci, V., Cerami, G.: The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. Arch. Ration. Mech. Anal. 114(1), 79–93 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benci, V., Cerami, G., Passaseo, D.: On the number of the positive solutions of some nonlinear elliptic problems, Nonlinear analysis, Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa 93-107 (1991)

  8. Cingolani, S., Tanaka, K.: Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam. 35, 1885–1924 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225–236 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Du Plessis, N.: Some theorems about the Riesz fractional integral. Trans. Am. Math. Soc. 80, 124–134 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  13. Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinburgh Sect. A 142, 1237–1262 (2012)

    Article  MATH  Google Scholar 

  14. Gao, F., Yang, M.: The Brézis-Nirenberg type critical problem for the nonlinear Choquard equation. Sci. China Math. 61(7), 1219–1242 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghimenti, M., Pagliardini, D.: Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains. Calc. Var. Partial. Differ. Equ. 58, 167 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. He, X., R\(\breve{a}\)dulescu, V.D.: Small linear perturbations of fractional Choquard equations with critical exponent. J. Differ. Equ. 282, 481–540 (2021)

  17. Ji, C., R\(\breve{a}\)dulescu, V.D.: Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well. J. Differ. Equ. 306, 251–279 (2022)

  18. Lieb, E.L., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)

  19. Liu, M., Tang, Z.: Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory. Discrete Contin. Dyn. Syst. 39(6), 3365–3398 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Mukherjee, T., Sreenadh, K.: Fractional Choquard equation with critical nonlinearities. Nonlinear Differ. Equ. Appl. 24, 63 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213, 587–628 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Servadei, R., Valdinoci, E.: The Brézis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)

    Article  MATH  Google Scholar 

  24. Siciliano, G.: Multiple positive solutions for a Schrödinger-Poisson-Slater system. J. Math. Anal. Appl. 365, 288–299 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Struwe, M.: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin (2008)

  26. Willem, M.: Minimax Theorems. Birckhäuser, Boston (1996)

    Book  MATH  Google Scholar 

  27. Xiang, M., R\(\breve{a}\)dulescu, V.D., Zhang, B.: A critical fractional Choquard-Kirchhoff problem with magnetic field. Commun. Contemp. Math. 21, 1850004 (2019)

  28. Yang, Z., Zhao, F.: Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Adv. Nonlinear Anal. 10, 732–774 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to show our sincere gratitude to the reviewers for their valuable suggestions, and especially for the relevant references suggested.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56126, Pisa, Italy

    Marco G. Ghimenti

  2. School of Mathematics, Liaoning Normal University, Dalian, 116029, China

    Min Liu

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Zhongwei Tang

Authors
  1. Marco G. Ghimenti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Min Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhongwei Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Min Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

M. Liu: Supported by NSFC (12101282), NSFLN (2022-BS-233), LJKZ0967 and 2021BSL004.

Z. Tang: Supported by NSFC (12071036,12126306).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghimenti, M.G., Liu, M. & Tang, Z. Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains. Nonlinear Differ. Equ. Appl. 30, 28 (2023). https://doi.org/10.1007/s00030-022-00838-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-022-00838-4

Keywords

Mathematics Subject Classification

Search

Navigation