Existence of solutions for a fractional Choquard-type equation in \(\mathbb {R}\) with critical exponential growth


In this paper, we study the following class of fractional Choquard-type equations

$$\begin{aligned} (-\Delta )^{1/2}u + u=\Big ( I_\mu *F(u)\Big )f(u), \quad x\in \mathbb {R}, \end{aligned}$$

where \((-\Delta )^{1/2}\) denotes the 1/2-Laplacian operator, \(I_{\mu }\) is the Riesz potential with \(0<\mu <1\), and F is the primitive function of f. We use variational methods and minimax estimates to study the existence of solutions when f has critical exponential growth in the sense of Trudinger–Moser inequality.

This is a preview of subscription content, access via your institution.


  1. 1.

    Adimurthi, Yadava S.L: Multiplicity results for semilinear elliptic equations in bounded domain of \(\mathbb{R}^2\) involving critical exponent. Ann. Sci. Norm. Super. Pisa 17, 481–504 (1990)

    MATH  Google Scholar 

  2. 2.

    Albuquerque, F.S.B., Ferreira, M.C., Severo, U.B.: Ground state solutions for a nonlocal equation in \(\mathbb{R}^{2}\) involving vanishing potentials and exponential critical growth, arXiv

  3. 3.

    Alves, C.O., 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. Equa. 261(3), 1933–1972 (2016)

    Article  Google Scholar 

  4. 4.

    Alves, C.O., Yang, M.: Existence of solutions for a nonlocal variational problem in \(\mathbb{R}^2\) with exponential critical growth. J. Convex Anal. 24, 1197–1215 (2017)

    MathSciNet  Google Scholar 

  5. 5.

    Arora, R., Giacomoni, J., Mukherjee, T., Sreenadh, K.: \(n\)-Kirchhoff-Choquard equations with exponential nonlinearity. Nonlinear Anal. 186, 113–144 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Caffarelli, L.: Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposium 7, Springer, Heidelberg, pp. 37–52 (2012)

  7. 7.

    Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb{R}^2\). Comm. Partial Differ. Equa. 17, 407–435 (1992)

    Article  Google Scholar 

  8. 8.

    Choquard, P., Stubbe, J.: The one-dimensional Schrödinger-Newton equations. Lett. Math. Phys. 81, 177–184 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Choquard, P., Stubbe, J., Vuffray, M.: Stationary solutions of the Schrödinger–Newton model—an ODE approach. Differ. Int. Equa. 21, 665–679 (2008)

    MATH  Google Scholar 

  10. 10.

    de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \(\mathbb{R}^{2}\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equa. 3, 139–153 (1995)

    Article  Google Scholar 

  11. 11.

    Del Pezzo, L.M., Quaas, A.: A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian. J. Differ. Equa. 263(1), 765–778 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    de Souza, M., Araújo, Y.L.: On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth. Math. Nachr. 289, 610–625 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Giacomoni, J., Mishra, P., Sreenadh, K.: Critical growth problems for \(1/2\)-Laplacian in \(\mathbb{R}\). Differ. Equ. Appl. 8, 295–317 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Iula, S., Maalaoui, A., Martinazzi, L.: A fractional Moser–Trudinger type inequality in one dimension and its critical points. Differ. Int. Equ. 29(5/6), 455–492 (2016)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kozono, H., Sato, T., Wadade, H.: Upper bound of the best constant of a Trudinger–Moser inequality and its application to a Gagliardo–Nirenberg inequality. Indiana Univ. Math. J. 55, 1951–1974 (2006)

    MathSciNet  Article  Google Scholar 

  17. 17.

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

  18. 18.

    Li, S., Shen, Z., Yang, M.: Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth. J. Math. Anal. Appl. 475, 1685–1713 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145 and 223–283 (1984)

  21. 21.

    Moroz, V., Van Schaftingen, J.: Existence of ground states for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367(9), 6557–6579 (2015)

    Article  Google Scholar 

  22. 22.

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

    MathSciNet  Article  Google Scholar 

  23. 23.

    Ozawa, T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259–269 (1995)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Pekar, S.: Untersuchungüber Die Elektronentheorie Der Kristalle. Akademie Verlag, Berlin (1954)

    MATH  Google Scholar 

  25. 25.

    Penrose, R.: On gravity role in quantum state reduction. Gen. Relativ. Gravitat. 28, 581600 (1996)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Sun, X., Zhu, A.: Multi-peak solutions for nonlinear Choquard equation in the plane. J. Math. Phys. 60, 1–18 (2019)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Takahashi, F.: Critical and subcritical fractional Trudinger–Moser-type inequalities on \(\mathbb{R}\). Adv. Nonlinear Anal. 1, 868–884 (2019)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Yang, M.: Semiclassical ground state solutions for a Choquard type equation in \(\mathbb{R}^2\) with critical exponential growth. ESAIM Control Optim. Calc. Var. 24, 177–209 (2018)

    MathSciNet  Article  Google Scholar 

Download references


The authors would like to express their sincere gratitude to the referees for carefully reading the manuscript and valuable comments and suggestions.

Author information



Corresponding author

Correspondence to Rodrigo Clemente.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Clemente, R., de Albuquerque, J.C. & Barboza, E. Existence of solutions for a fractional Choquard-type equation in \(\mathbb {R}\) with critical exponential growth. Z. Angew. Math. Phys. 72, 16 (2021). https://doi.org/10.1007/s00033-020-01447-w

Download citation

Mathematics Subject Classification

  • 35J20
  • 35J60
  • 35R11


  • Fractional Choquard-type equation
  • Critical exponential growth
  • Trudinger–Moser inequality