Skip to main content

Advertisement

SpringerLink
Log in

On some singular semilinear elliptic equation with doubly exponential growth at infinity in \({\mathbb {R}}^2 \)

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

In this paper, we prove the existence of at least two solutions to some semilinear elliptic equation defined in the whole euclidean space \( {\mathbb {R}}^2.\) The nonlinearity consists of two terms: one presents a singularity at the origin and the other has a doubly exponential behavior at infinity. This new exponential behavior is governed by a recently established weighted inequality of Trudinger–Moser type. A combination of perturbation arguments and variational tools is employed to obtain our multiplicity result.

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. Aouaoui, S.: A new Trudinger–Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space \( {\mathbb{R}}^2\). Arch. Math. 114, 199–214 (2020)

    Article  MathSciNet  Google Scholar 

  2. Aouaoui, S.: On some quasilinear equation with critical exponential growth at infinity and a singular behavior at the origin. J. Elliptic Parabol Equ. 4, 27–50 (2018)

    Article  MathSciNet  Google Scholar 

  3. Calanchi, M., Ruf, B.: On Trudinger-Moser type inequalities with logarithmic weights. J. Differ. Equ. 258, 1967–1989 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cannino, A., Sciunzi, B., Trombetta, A.: Existence and uniqueness for p-Laplace equations involving singular nonlinearities. NoDEA Nonlinear Differ. Equ. Appl. 23, 8 (2016). https://doi.org/10.1007/s00030-016-0361-6

    Article  MathSciNet  MATH  Google Scholar 

  5. Carl, S., Perera, K.: Generalized solutions of singular p-Laplacian problems in \( {\mathbb{R}}^N\). Nonlinear Stud. 18, 113–124 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Carmona, J., Leonori, T., Lopez-Martinez, S., Martinez-Aparacio, P.J.: Quasilinear elliptic problems with singular and homogeneous lower order terms. Nonlinear Anal. 179, 105–130 (2019)

    Article  MathSciNet  Google Scholar 

  7. Dall’Aglio, A., De Cicco, V., Giachetti, D., Puel, J.-P.: Existence of bounded solutions for nonlinear elliptic equations in unbounded domains. NoDEA Nonlinear Differ. Equ. Appl. 11, 431–450 (2004)

    Article  MathSciNet  Google Scholar 

  8. Drábek, P., Sankar, L.: Singular quasilinear elliptic problems on unbounded domains. Nonlinear Anal. 109, 148–155 (2014)

    Article  MathSciNet  Google Scholar 

  9. do Ó, J.M., da Silva, E.: Quasilinear elliptic equations with singular nonlinearity. Adv. Nonlinear Stud. 16 (2016) 363-380

  10. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    Book  Google Scholar 

  11. Goyal, S., Sreenadh, K.: On multiplicity of positive solutions for N-Laplacian with singular and critical nonlinearity. Complex Var. Elliptic Equ. 59, 1636–1649 (2014)

    Article  MathSciNet  Google Scholar 

  12. Lei, C.Y., Liao, J.F., Tang, C.L.: Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents. J. Math. Anal. Appl. 421, 521–538 (2015)

    Article  MathSciNet  Google Scholar 

  13. Li, Q.W., Gao, W.J.: Existence of weak solutions to a class of singular elliptic equations. Mediterr. J. Math. 13, 4917–4927 (2016)

    Article  MathSciNet  Google Scholar 

  14. Montenegro, M.: Strong maximum principles for supersolutions of quasilinear elliptic equations. Nonlinear Anal. 37, 431–448 (1999)

    Article  MathSciNet  Google Scholar 

  15. Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)

    Article  MathSciNet  Google Scholar 

  16. Prashanth, S., Tiwari, S., Sreenadh, K.: Very singular problems with critical nonlinearities in two dimensions. Commun. Contemp. Math. 1650067 (2017)

  17. Rǎdulescu, V., Pucci, P.: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. Ser. IX(3), 543–584 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Rǎdulescu, V., Ghergu, M.: Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, vol. 37. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  19. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Kairouan, Department of Mathematics, High Institute of Applied Mathematics and Informatics of Kairouan, Avenue Assad Iben Fourat, 3100, Kairouan, Tunisia

    Sami Aouaoui

  2. University of Sousse, High School of Sciences and Technology of Hammam Sousse, Rue Lamine ElAbbessi, 4011, Hammam Sousse, Tunisia

    Ala Eddine Bahrouni

  3. Laboratory of Algebra,Number Theory and Nonlinear Analysis LR18ES15, Faculty of Sciences of Monastir, Rue de l’environnement, 5019, Monastir, Tunisia

    Ala Eddine Bahrouni

Authors
  1. Sami Aouaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ala Eddine Bahrouni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sami Aouaoui.

Ethics declarations

Conflict of intrest

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aouaoui, S., Bahrouni, A.E. On some singular semilinear elliptic equation with doubly exponential growth at infinity in \({\mathbb {R}}^2 \). Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00706-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11587-022-00706-4

Keywords

Mathematics Subject Classification

Search

Navigation