Skip to main content
SpringerLink
Log in

On Some Weighted 1-Laplacian Problem on \( {\mathbb {R}}^N \) with Singular Behavior at the Origin

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this work, we prove the existence of a nontrivial solution to a quasilinear elliptic problem defined on the whole Euclidean space \( {\mathbb {R}}^N,\ N \ge 2, \) and involving a weighted 1-Laplacian operator. The nonlinear term has a singular behavior at the origin. This solution is obtained through an approximation technique, which consists in considering the problem with the 1-Laplacian operator as a limit of a family of problems with the p-Laplacian operators when \(p \rightarrow 1^+.\) For that aim, a new version of Anzellotti’s \( L^{\infty }-\)divergence−measure pairing theory is established and new arguments are used.

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. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180, 347–403 (2001)

    Article  MathSciNet  Google Scholar 

  2. Andreu, F., Caselles, V., Mazón, J.M.: Parabolic quasilinear equations minimizing linear growth functionals, Progress in Mathematics, 223. Birkhäuser Verlag, Basel (2004)

    Book  Google Scholar 

  3. Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135(1), 293–318 (1983)

    Article  MathSciNet  Google Scholar 

  4. Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and \(BV\) spaces applications to PDEs and optimization, SIAM (2006) Philadelphia, USA

  5. Baldi, A.: Weighted \(BV\) functions. Houston J. Math 27(3), 683–705 (2001)

    MathSciNet  Google Scholar 

  6. Chata, J.C.O., Pimenta, M.T.O.: León, Sergio Segura de: Anisotropic \(1-\)Laplacian problems with unbounded weights. NoDEA Nonlinear Diff. Equ. Appl. 28, 57 (2021). https://doi.org/10.1007/s00030-021-00717-4

    Article  Google Scholar 

  7. Chen, G.Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147, 89–118 (1999)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)

    Article  MathSciNet  Google Scholar 

  9. Cicalese, M., Trombetti, C.: Asymptotic behaviour of solutionsto \(p-\)Laplacian equation. Asymptot. Anal. 35, 27–40 (2003)

    MathSciNet  Google Scholar 

  10. Crasta, G., De Cicco, V.: An extension of the pairing theory between divergence-measure fileds ABD BV functions and applications. J. Funct. Anal. 276, 2605–2635 (2019)

    Article  MathSciNet  Google Scholar 

  11. Crasta, G., De Cicco, V.: Anzellotti’s pairing theory and the Gauss-Green theorem. Adv. Math. 343, 935–970 (2019)

    Article  MathSciNet  Google Scholar 

  12. Crasta, G., De Cicco, V., Malusa, A.: Pairings between bounded divergence-measure vector fields and BV functions. Adv. Calc. Var. 4, 787–810 (2022)

    Article  MathSciNet  Google Scholar 

  13. De Cicco, V., Giachetti, D., Oliva, F., Pettita, F.: Dirichlet problems for singular elliptic equations with general nonlinearities. Calc. Var. Partial Diff. Equ. 58, 4 (2019)

    Article  MathSciNet  Google Scholar 

  14. De Cicco, V., Giachetti, D., Segura de León, S.: Elliptic problems involving the \(1-\)Laplacian and a singular lower order term. J. Lond. Math. Soc. 2(99), 349–376 (2019)

    Article  MathSciNet  Google Scholar 

  15. Demengel, F.: Théorèmes d’existence pour des équations avec l’opérateur “\(1-\)laplacien’’, première valeur propre pour \(\Delta _{1}\). C. R. Math. Acad. Sci. Paris 334(12), 1071–1076 (2002)

    Article  MathSciNet  Google Scholar 

  16. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  Google Scholar 

  17. El Hadfi, Y., El Hichami, M.: On 1-Laplacian elliptic problems involving a singular term and an \(L^1-\)data. J. Elliptic Parabol Equ. (2023). https://doi.org/10.1007/s41808-023-00210-2

    Article  Google Scholar 

  18. Figueiredo, G.M., Pimenta, M.T.O.: Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions. NoDEA Nonlinear Diff. Equ. Appl. (2018). https://doi.org/10.1007/s00030-018-0538-2

    Article  MathSciNet  Google Scholar 

  19. Figueiredo, G.M., Pimenta, M.T.O.: Strauss and Lions Type Results in \(BV({\mathbb{R} }^N)\) with an application to an \(1-\)Laplacian Problem. Milan J. Math. 86, 15–30 (2018)

    Article  MathSciNet  Google Scholar 

  20. Figueiredo, G.M., Pimenta, M.T.O.: Existence of bounded variation solution for a \(1-\)Laplacian problem with vanishing potentials. J. Math. Anal. Appl. 459, 861–878 (2018)

    Article  MathSciNet  Google Scholar 

  21. Figueiredo, G.M., Pimenta, M.T.O.: Sub-supersolution method for a quasilinear elliptic problem involving the \(1-\)laplacian operator and a gradient term. J. Funct. Anal. 278, 108325 (2020)

    Article  MathSciNet  Google Scholar 

  22. Figueiredo, G., Pimenta, M.T.O.: Nodal solutions to quasilinear elliptic problems involving the \(1-\)Laplacian operator via variational and approximation methods. Indiana Univ. Math. J. 71, 439–462 (2022)

    Article  MathSciNet  Google Scholar 

  23. Filippucci, R., Pucci, P., Radulescu, V.: Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary condition. Comm. Partial Diff. Equ. 33, 706–717 (2008)

    Article  MathSciNet  Google Scholar 

  24. Giachetti, D., Oliva, F., Petitta, F.: \(1-\)Laplacian type problems with strongly singular nonlinearities and gradient terms. Commun. Contemp. Math. 24(10), 2150081 (2022)

    Article  MathSciNet  Google Scholar 

  25. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Diff. Geom. 59, 353–438 (2001)

    MathSciNet  Google Scholar 

  26. Kavian, O.: Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Springer-Verlag, Paris (1993)

    Google Scholar 

  27. Kilpeläinen, T.: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn.Ser. A I Math 19, 95–113 (1994)

    MathSciNet  Google Scholar 

  28. Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)

    MathSciNet  Google Scholar 

  29. Latorre, M., Oliva, F., Petitta, F., Segura de León, S.: The Dirichlet problem for the \(1-\)Laplacian with a general singular term and \(L^1-\)data. Nonlinearity 34, 1791–1816 (2021)

    Article  MathSciNet  Google Scholar 

  30. Mazón, J.M., Segura de León, S.: The Dirichlet problem of a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow. Adv. Calc. Var. 6, 123–164 (2013)

    Article  MathSciNet  Google Scholar 

  31. Mercaldo, A., Segura de León, S., Trombetti, C.: On the solutions to \(1-\)Laplacian equation with \(L^1\) data. J. Funct. Anal. 256, 2387–2416 (2009)

    Article  MathSciNet  Google Scholar 

  32. Mercaldo, A., Rossi, J., Segura de León, S., Trombetti, C.: Anisotropic \(p, q-\)Laplacian equations when \(p\) goes to 1. Nonlinear Anal. 73(11), 3546–3560 (2010)

    Article  MathSciNet  Google Scholar 

  33. Mercaldo, A., Rossi, J.D., Segura de León, S., Trombetti, C.: Behaviour of \(p-\)Laplacian problems with Neumann boundary conditions when \(p\) goes to 1. Commun. Pure Appl. Anal. 12(1), 253–267 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Pimenta, M.T.O., Gonzaga, A.S.: Symmetry and symmetry breaking for Henon type problems involving the \(1-\)Laplacian operator. Commun. Contemp. Math. (2022). https://doi.org/10.1142/S0219199722500213

    Article  Google Scholar 

  36. Pimenta, M.T.O., Santos, G.C.G., Santos Júnior, J.R.: On a quasilinear elliptic problem involving the \(1-\)Laplacian operator and a discontinuous nonlinearity. Proc. Roy. Soc. Edinburgh Sect. A Math. (2022). https://doi.org/10.1017/prm.2022.86

    Article  Google Scholar 

  37. Salas, A.M., Segura de León, S.: Elliptic equations involving the \(1-\)Laplacian and a subcritical source term. Nonlinear Anal. 168, 50–66 (2018)

    Article  MathSciNet  Google Scholar 

  38. Simon, J.: Régularité de la solution d’une équation non linéaire dans \(\mathbb{R^N},\) (French) Journées d’Analyse Non Linéaire (Proc. Conf, Besançon, 1977), pp. 205-227, Lecture Notes in Math, 665, Springer, Berlin,1978

  39. Zhou, F., Shen, Z.: Existence of a radial solution to a \(1-\)Laplacian problem in \({\mathbb{R} }^N\). Appl. Math. Lett. 118, 107138 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous referee for his/her careful reading of the manuscript and his/her insightful and constructive remarks and comments that helped to clarify the content and improve the presentation of the results in this paper considerably.

Author information

Authors and Affiliations

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

    Sami Aouaoui

  2. Faculty of Sciences of Monastir, University of Monastir, Avenue de l’environnement, 5019, Monastir, Tunisia

    Mariem Dhifet

Authors
  1. Sami Aouaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mariem Dhifet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sami Aouaoui.

Ethics declarations

Conflict of interest

This work does not have any conflict of interest.

Additional information

Communicated by Rosihan M. Ali.

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

Aouaoui, S., Dhifet, M. On Some Weighted 1-Laplacian Problem on \( {\mathbb {R}}^N \) with Singular Behavior at the Origin. Bull. Malays. Math. Sci. Soc. 47, 20 (2024). https://doi.org/10.1007/s40840-023-01622-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-023-01622-y

Keywords

Mathematics Subject Classification

Search

Navigation