Skip to main content
SpringerLink
Log in

On nonlinear perturbations of a periodic integrodifferential Kirchhoff equation with critical exponential growth

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we investigate the existence of solutions for a class of integrodifferential Kirchhoff equations. These equations involve a nonlocal operator with a measurable kernel that satisfies “structural properties” that are more general than the standard kernel of the fractional Laplacian operator. Additionally, the potential can be periodic or asymptotically periodic, and the nonlinear term exhibits critical exponential growth in the sense of Trudinger–Moser inequality. To guarantee the existence of solutions, we employ variational methods, specifically the mountain-pass theorem. In this context, it is important to emphasize that we have additional difficulties due to the lack of compactness in our problem, because we deal with critical growth nonlinearities in unbounded domains. Moreover, the Kirchhoff term adds complexity to the problem, as it requires suitable calculations for control the estimate the minimax level, representing the main challenge in this work. Finally, we consider two different approaches to estimate the minimax level. The first approach is based on a hypothesis proposed by D. M. Cao, while the second one involves a slightly weaker assumption addressed by Adimurthi and Miyagaki.

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

    Article  MathSciNet  Google Scholar 

  2. Kirchhoff, G.: Mechanik. Teubner, L., Germany. (1883)

  3. Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997)

    Article  MathSciNet  Google Scholar 

  4. Alves, C.O., Corrêa, F.: On existence of solutions for a class of problem involving a nonlinear operator. Appl. Nonlinear Anal. 8, 43–56 (2001)

    MathSciNet  Google Scholar 

  5. de Albuquerque, J. C., Araújo, Y. L., Clemente, R. G.: Existence of bound and ground states for a class of Kirchhoff–Schrödinger equations involving critical Trudinger–Moser growth. Math. Nachr. 289, 610–625 (2016)

  6. Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)

    Article  MathSciNet  Google Scholar 

  7. Liu, Z., Squassina, M., Zhang, J.: Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. NoDEA Nonlinear Differ. Equ. Appl. 24, 24–50 (2017)

    Article  MathSciNet  Google Scholar 

  8. Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)

    Article  MathSciNet  Google Scholar 

  9. Fiscella, A., Pucci, P.: \(p\)-fractional Kirchhoff equations involving critical nonlinearities. Nonlinear Anal. Real World Appl. 35, 350–378 (2017)

    Article  MathSciNet  Google Scholar 

  10. Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MathSciNet  Google Scholar 

  11. Ambrosio, V., Isernia, T.: A multiplicity result for a fractional Kirchhoff equation in \({\mathbb{R}}^N\) with a general nonlinearity. Commun. Contemp. Math. 20 (2018)

  12. Berestycki, H., Lions, P.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Araújo, Y. L., Barboza, E., de Carvalho, G.: On nonlinear perturbations of a periodic integrodifferential equation with critical exponential growth. Appl. Anal. (2021)

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

    Article  MathSciNet  Google Scholar 

  16. Iula, S.: A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one. Atti. Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28, 871–884 (2017)

    MathSciNet  Google Scholar 

  17. 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)

    Article  MathSciNet  Google Scholar 

  18. Takahashi, F.: Critical and subcritical fractional Trudinger–Moser-type inequalities on \(\mathbb{R} \). Adv. Nonlinear Anal. 8, 868–884 (2019)

    Article  MathSciNet  Google Scholar 

  19. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)

  20. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)

    MathSciNet  Google Scholar 

  21. Miyagaki, O. H., Pucci, P.: Nonlocal Kirchhoff problems with Trudinger–Moser critical nonlinearities. Nonlinear Differ. Equ. Appl. 26 (2019). https://doi.org/10.1007/s00030-019-0574-6

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

    Article  Google Scholar 

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

  24. 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. Equ. 3, 139–153 (1995)

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  26. de Sousa, K.C.V., Tavares, L.S.: On an indefinite nonhomogeneous equation with critical exponential growth. Partial Differ. Equ. Appl. 4, 28 (2023)

    Article  MathSciNet  Google Scholar 

  27. 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)

    Article  MathSciNet  Google Scholar 

  28. Perera, K., Squassina, M.: Bifurcation results for problems with fractional Trudinger–Moser nonlinearity. Discrete Contin. Dyn. Syst. Ser. S 11, 561–576 (2018)

    MathSciNet  Google Scholar 

  29. Adimurthi, A.: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the \(n\)-Laplacian. Ann. Sc. Norm. Super. Pisa 17, 393–413 (1990)

  30. Lins, H.F., Silva, E.A.B.: Quasilinear asymptotically periodic elliptic equations with critical growth. Nonlinear Anal. 71, 2890–2905 (2009)

    Article  MathSciNet  Google Scholar 

  31. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, New York (1989)

    Book  Google Scholar 

  32. Lions, P.L.: Symétrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)

    Article  Google Scholar 

  33. Pei, R.: Fractional \(p\)-Laplacian equations with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition. Mediterr. J. Math. 15, 5 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal Rural de Pernambuco, Rua Dom Manuel de Medeiros, Recife, Pernambuco, Brazil

    Eudes Barboza, Yane Araújo & Gilson de Carvalho

Authors
  1. Eudes Barboza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yane Araújo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gilson de Carvalho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eudes Barboza.

Additional information

Publisher's Note

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

This work was supported in part by FACEPE – APQ–1094-1.01/22.

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

Barboza, E., Araújo, Y. & Carvalho, G.d. On nonlinear perturbations of a periodic integrodifferential Kirchhoff equation with critical exponential growth. Z. Angew. Math. Phys. 74, 225 (2023). https://doi.org/10.1007/s00033-023-02117-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-02117-3

Keywords

Mathematics Subject Classification

Search

Navigation