Skip to main content
Springer Nature Link
Log in

Multiplicity and Concentration Properties for Fractional Choquard Equations with Exponential Growth

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

This article deals with the fractional Choquard equation involving exponential growth as follows

$$ \varepsilon ^{N}(-\Delta )_{p}^{s}u+Z(x)|u|^{p-2}u =\varepsilon ^{\mu -N}\left[ \frac{1}{|x|^{\mu }}*H(u)\right] h(u)\; \ \ \text {in}\; \mathbb R^{N},$$

where \(\varepsilon \) a positive parameter, \(s\in (0,1)\), \(\mu \in (0, N)\) and \((-\Delta )_{p}^{s}\) is a fractional p-Laplace operator with \(p=\frac{N}{s}\ge 2.\) The function h is only continuous and has exponential growth. In addition, the potential function Z satisfies some appropriate conditions. We use the Trudinger–Moser inequality to deal with the function h involving exponential growth. Together with the Ljusternik–Schnirelmann category theory and variational method, the multiplicity and concentration behavior of positive solutions are obtained for the above problem. As far as we know, our results seem to be new for the fractional \(\frac{N}{s}\)-Laplacian Choquard equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Alves, C., Cassani, D., Tarsi, C., Yang, M.: Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \(\mathbb{R} ^2\). J. Differ. Equ. 261, 1933–1972 (2016)

    Article  Google Scholar 

  2. Alves, C.: Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Proc. R. Soc. Edinb. A 146, 23–58 (2016)

    Article  MathSciNet  Google Scholar 

  3. Alves, C., Gao, F., Squassina, M., Yang, M.: Singularly perturbed critical Choquard equations. J. Differ. Equ. 263, 3943–3988 (2017)

    Article  MathSciNet  Google Scholar 

  4. Ambrosio, V., Isernia, T.: Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional \(p\)-Laplace. Discrete Contin. Dyn. Syst. 38, 5835–5881 (2018)

    Article  MathSciNet  Google Scholar 

  5. Ambrosio, V.: Multiplicity and concentration results for a fractional Choquard equation via penalization method. Potential Anal. 50, 55–82 (2019)

    Article  MathSciNet  Google Scholar 

  6. Ambrosio, V.: On the multiplicity and concentration of positive solutions for a \(p\)-fractional Choquard equation in \(\mathbb{R} ^N\). Comput. Appl. Math. 78, 2593–2617 (2019)

    Article  MathSciNet  Google Scholar 

  7. Ambrosio, V.: Nonlinear fractional Schrödinger equations in \(\mathbb{R}^{N}.\) In: Frontiers in Elliptic and Parabolic Problems. Birkhäuser/Springer, Cham (2021)

  8. Bisci, G., Thin, N.V., Vilasi, L.: On a class of nonlocal Schrödinger equations with exponential growth. Adv. Differ. Equ. 27, 571–610 (2022)

    Google Scholar 

  9. Caffarelli, L., Silvesytre, L.: An extension problems related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)

    Article  MathSciNet  Google Scholar 

  10. Chen, S., Li, L., Yang, Z.: Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent. RACSAM 114, 33 (2020)

    Article  MathSciNet  Google Scholar 

  11. Chen, S., Shu, M., Tang, X., Wen, L.: Planar Schrödinger–Poisson system with critical exponential growth in the zero mass case. J. Differ. Equ. 327, 448–480 (2022)

    Article  Google Scholar 

  12. 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  Google Scholar 

  13. Clemente, R., Albuquerque, J.C.D., Barboza, E.: Existence of solutions for a fractional Choquard-type equation in \(\mathbb{R} ^N\) with critical exponential growth. Z. Angew. Math. Phys. 72, 16 (2021)

    Article  Google Scholar 

  14. de Böer, E., Miyagaki, O.H.: Existence and multiplicity of solutions for the fractional \(p\)-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth. J. Math. Phys. 62, 051507 (2021)

    Article  MathSciNet  Google Scholar 

  15. Del Pezzo, L., Quaas, A.: A Hopf’s lemma and a strong minimum principle for the fractional \(p\)-Laplacian. J. Differ. Equ. 263, 765–778 (2017)

    Article  MathSciNet  Google Scholar 

  16. Figueiredo, G., Bisci, G.M., Servadei, R.: The effect of the domain topology on the number of solutions of fractional Laplace problems. Calc. Var. Partial Differ. Equ. 57, 1–24 (2018)

    Article  MathSciNet  Google Scholar 

  17. Floer, A., Weinstein, A.: Non spreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  Google Scholar 

  18. Fröhlich, H.: Theory of electrical breakdown in ionic crystal. Proc. R. Soc. A 160, 230–241 (1937)

    Google Scholar 

  19. Fröhlich, H.: Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954)

    Article  Google Scholar 

  20. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoam. 32, 1353–1392 (2016)

    Article  MathSciNet  Google Scholar 

  21. Jones, K.: Newtonian quantum gravity. Aust. J. Phys. 48, 1055–1081 (1995)

    Article  Google Scholar 

  22. Lia, Q., Yang, Z.: Multiple solutions for a class of fractional quasi-linear equations with critical exponential growth in \(\mathbb{R} ^N,\). Complex Var. Elliptic Equ. 61, 969–983 (2016)

    Article  MathSciNet  Google Scholar 

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

  24. Lieb, E., Loss, M.: Analysis: Graduate Studies in Mathematics. AMS, Providence (2001)

    Google Scholar 

  25. Lions, P.: The Choquard equation and related questions. Nonlinear Anal. Theory Methods Appl. 4, 1063–1072 (1980)

    Article  MathSciNet  Google Scholar 

  26. Liu, Z., Rădulescu, V., Tang, C., Zhang, J.: Another look at planar Schrödinger–Newton systems. J. Differ. Equ. 328, 65–104 (2022)

    Article  Google Scholar 

  27. Meng, Y., He, X.: Multiplicity of concentrating solutions for Choquard equation with critical growth. J. Geom. Anal. 33, 78 (2023)

    Article  MathSciNet  Google Scholar 

  28. Moroz, V., Schaftingen, J.V.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19, 773–813 (2017)

    Article  MathSciNet  Google Scholar 

  29. Moroz, I., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger–Newton equations. Class. Quantum Gravity 15, 2733–2742 (1998)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Papageorgiou, N.S., Zhang, J., Zhang, W.: Global existence and multiplicity of solutions for nonlinear singular eigenvalue problems. Discrete Contin. Dyn. Syst. S (2024). https://doi.org/10.3934/dcdss.2024018

    Article  Google Scholar 

  32. Papageorgiou, N.S., Rădulescu, V.D., Zhang, W.: Global existence and multiplicity for nonlinear Robin eigenvalue problems. Results Math. 78, 133 (2023)

    Article  MathSciNet  Google Scholar 

  33. Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle, p. 2. Akademie Verlag, Berlin (1954)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R} ^N\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)

    Article  Google Scholar 

  36. Qin, D., Rădulescu, V.D., Tang, X.H.: Ground states and geometrically distinct solutions for periodic Choquard–Pekar equations. J. Differ. Equ. 275, 652–683 (2021)

    Article  MathSciNet  Google Scholar 

  37. Rabinowitz, P.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)

    Article  MathSciNet  Google Scholar 

  38. Singh, G.: Nonlocal perturbations of the fractional Choquard equation. Adv. Numer. Anal. 8, 694–706 (2017)

    MathSciNet  Google Scholar 

  39. Su, Y., Wang, L., Chen, H., Liu, S.: Multiplicity and concentration results for fractional Choquard equations: doubly critical case. Nonlinear Anal. 198, 111872 (2020)

    Article  MathSciNet  Google Scholar 

  40. Szulkin, A., Weth, T.: The method of Nehari manifold. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, pp. 597–632. International Press, Boston (2010)

    Google Scholar 

  41. Thin, N.: Singular Trudinger–Moser inequality and fractional \(p\)-Laplace equation in \(\mathbb{R} ^N\). Nonlinear Anal. 196, 111756 (2020)

    Article  MathSciNet  Google Scholar 

  42. Thin, N.: Multiplicity and concentration of solutions to a fractional \(p\)-Laplace problem with exponential growth. Ann. Fenn. Math. 47, 603–639 (2022)

    Article  MathSciNet  Google Scholar 

  43. Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)

    Book  Google Scholar 

  44. Ye, H.: The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in \(\mathbb{R} ^N\). J. Math. Anal. Appl. 431, 935–954 (2015)

    Article  MathSciNet  Google Scholar 

  45. 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  Google Scholar 

  46. Yuan, S., Tang, X., Zhang, J., Zhang, L.: Semiclassical states of fractional Choquard equations with exponential critical growth. J. Geom. Anal. 32, 290 (2022)

    Article  MathSciNet  Google Scholar 

  47. Yuan, S., Rădulescu, V.D., Tang, X., Zhang, L.: Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction. Forum Math. 36, 783–810 (2024)

    Article  MathSciNet  Google Scholar 

  48. Zhang, C.: Trudinger–Moser inequalities in Fractional Sobolev–Slobodeckij spaces and multiplicity of weak solutions to the Fractional-Laplacian equation. Adv. Nonlinear Stud. 19, 197–217 (2019)

    Article  MathSciNet  Google Scholar 

  49. Zhang, H., Zhang, F.: Multiplicity and concentration of solutions for Choquard equations with critical growth. J. Math. Anal. Appl. 481, 123457 (2020)

    Article  MathSciNet  Google Scholar 

  50. Zhang, J., Lü, W., Lou, Z.: Multiplicity and concentration behavior of solutions of the critical Choquard equation. Appl. Anal. 100, 167–190 (2021)

    Article  MathSciNet  Google Scholar 

  51. Zhang, J., Zhang, Y.: An infinite sequence of localized semiclassical states for nonlinear Maxwell–Dirac system. J. Geom. Anal. 34, 277 (2024)

    Article  MathSciNet  Google Scholar 

  52. Zhang, X., Sun, X., Liang, S., Thin Nguyen, V.: Existence and concentration of solutions to a Choquard equation involving fractional \(p\)-Laplacian via penalization method. J. Geom. Anal. 34, 90 (2024)

    Article  Google Scholar 

Download references

Acknowledgements

S. Shi was supported by NSFC (Grant No. 12271203) and Science and Technology Development Project of Jilin Province (Grant No. YDZJ202101ZYTS141). Thin Van Nguyen is supported by Thai Nguyen University of Education.

Author information

Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130022, Jilin, People’s Republic of China

    Shuaishuai Liang & Shaoyun Shi

  2. Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen City, Thai Nguyen, Vietnam

    Thin Van Nguyen

Authors
  1. Shuaishuai Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaoyun Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thin Van Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thin Van Nguyen.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

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

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

Liang, S., Shi, S. & Van Nguyen, T. Multiplicity and Concentration Properties for Fractional Choquard Equations with Exponential Growth. J Geom Anal 34, 367 (2024). https://doi.org/10.1007/s12220-024-01815-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-024-01815-2

Keywords

Mathematics Subject Classification

Search

Navigation