Skip to main content
SpringerLink
Log in

Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

In this paper, we study a class of nonlinear elliptic problems whose model is the following

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&-\textrm{div}\Big (b(u) |\nabla u|^{p-2}\nabla u\Big )=f\Big (1+\frac{1}{|u|^\gamma }\Big )\ \ \textrm{in}\ \Omega , \\ {}&u=0\ \ \textrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N (N\ge 2)\), \(\gamma > 0\), b is a positive continuous function which blows up for a finite value of the unknown u. We will prove existence and uniqueness of a renormalized nonnegative solution in the case where the nonnegative source f belongs to \(L^1(\Omega )\).

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

Data availability

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

References

  1. Blanchard, D., Guibé, O., Redwane, H.: Nonlinear equations with unbounded heat conduction and integrable data. Annali di Matematica 187(3), 405–433 (2008)

    Article  MathSciNet  Google Scholar 

  2. Blanchard, D., Redwane, H.: Quasilinear diffusion problems with singular coefficients with respect to the unknown. Proc. R. Soc. Edinb. A 132(5), 1105–1132 (2002)

    Article  MathSciNet  Google Scholar 

  3. Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. Partial Differ. Equ. 37(3–4), 363–380 (2010)

    Article  MathSciNet  Google Scholar 

  4. Boccardo, L., Murat, F., Puel, J.-P.: Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152, 183–196 (1988)

    Article  MathSciNet  Google Scholar 

  5. Bouhlal, A., Igbida, J.: Existence and regularity of solutions for unbounded elliptic equations with singular nonlinearities. Int. J. Differ. Equ. 2021, 1–9 (2021)

    MathSciNet  Google Scholar 

  6. Canino, A., Sciunzi, B., Trombetta, A.: Existence and uniqueness for p-Laplace equations involving singular nonlinearities. NoDEA Nonlinear Differ. Equ. Appl. 23, 1–18 (2016)

    Article  MathSciNet  Google Scholar 

  7. Carmona, J., Martínez-Aparicio, P.J.: A singular semilinear elliptic equation with a variable exponent. Adv. Nonlinear Stud. 16, 491–498 (2016)

    Article  MathSciNet  Google Scholar 

  8. Casado-Díaz, J., Murat, F.: Semilinear problems with right-hand sides singular at \(u =0\) which change sign. Annales de l’Institut Henri Poincaré- Analyse nonlinéaire 38, 877–909 (2021)

    Article  MathSciNet  Google Scholar 

  9. De Cave, L.M., Oliva, F.: Elliptic equations with general singular lower order term and measure data. Nonlinear Anal. 128, 391–411 (2015)

    Article  MathSciNet  Google Scholar 

  10. De Cave, L.M., Oliva, F.: On the regularizing effect of some absorption and singular lower order terms in classical Dirichlet problems with \(L^1\) data. J. Elliptic Parabol. Equ. 2(1–2), 73–85 (2016)

    Article  MathSciNet  Google Scholar 

  11. De Cave, L.M., Durastanti, R., Oliva, F.: Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data. NoDEA Nonlinear Differ. Equ. Appl. 547, 18–25 (2018)

    Article  MathSciNet  Google Scholar 

  12. Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193–222 (1977)

    Article  MathSciNet  Google Scholar 

  13. Croce, G.: An elliptic problem with two singularities. Asymptot. Anal. 78(1–2), 1–10 (2012)

    MathSciNet  Google Scholar 

  14. Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28(4), 741–808 (1999)

    MathSciNet  Google Scholar 

  15. Dávila, J., Montenegro, M.: Positive versus free boundary solutions to a singular elliptic equation. J. Anal. Math. 90, 303–335 (2003)

    Article  MathSciNet  Google Scholar 

  16. Della Pietra, F., Di Blasio, G.: Existence results for nonlinear elliptic problems with unbounded coefficient. Nonlinear Anal. 71(1–2), 72–87 (2009)

    Article  MathSciNet  Google Scholar 

  17. Diaz, J.I., Morel, J.-M., Oswald, L.: An elliptic equation with singular nonlinearity. Commun. Partial Differ. Equ. 12, 1333–1344 (1987)

    Article  MathSciNet  Google Scholar 

  18. Durastanti, R., Oliva, F.: The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity. Adv. Differ. Equ. 29(5–6), 339–388 (2024)

    MathSciNet  Google Scholar 

  19. Feo, F., Guibé, O.: Nonlinear problems with unbounded coefficients and \(L^1\) data. Nonlinear Differ. Equ. Appl. 27, 49 (2020)

    Article  Google Scholar 

  20. Garain, P.: On a degenerate singular elliptic problem. Mathematische Nachrichten 295(7), 1354–1377 (2022)

    Article  MathSciNet  Google Scholar 

  21. García Vázquez, C., Ortegón Gallego, F.: Sur un problème elliptique nonlinéaire avec diffusion singulière et second membre dans \(L^1(\Omega )\). C. R. Acad. Sci. Paris Sér. I Math. 332(2), 145–150 (2001)

    Article  MathSciNet  Google Scholar 

  22. García Vázquez, C., Ortegón Gallego, F.: An elliptic equation with blowing-up diffusion and data in \(L^1(\Omega )\), existence and uniqueness. Math. Models Methods Appl. Sci. 13(9), 1351–1377 (2003)

    Article  MathSciNet  Google Scholar 

  23. Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: A semilinear elliptic equation with a mild singularity at \(u = 0\): existence and homogenization. J. Math. Pures Appl. 107, 41–77 (2017)

    Article  MathSciNet  Google Scholar 

  24. Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991)

    Article  MathSciNet  Google Scholar 

  25. Leray, J., Lions, J.L.: Quelques résultats de Visik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder. Bull. Soc. Math. Fr. 93, 97–107 (1965)

    Article  Google Scholar 

  26. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaire. Dunod et Gauthier-Villars, Paris (1969)

    Google Scholar 

  27. Marah, A., Redwane, H., Zaki, K.: Nonlinear elliptic equations with unbounded coefficient and singular lower order term. J. Fixed Point Theory Appl. 22, 68 (2020)

    Article  MathSciNet  Google Scholar 

  28. Marah, A., Redwane, H.: On nonlinear elliptic equations with singular lower order term. Bull. Korean Math. Soc. 58, 2 (2021)

    MathSciNet  Google Scholar 

  29. Murat, F.: Soluciones renormalizadas de EDP elipticas non lineales. Cours à l’université de Séville, Laboratoire d\(^{\prime }\)Analyse Numérique, Publication R93023, Paris VI (1993)

  30. Murat, F.: Equations elliptiques non linéaires avec second membre \(L^1\) ou mesure. In: Compte Rendus du 26 ème Congrès d’Analyse Numérique, les Karellis (1994)

  31. Oliva, F., Petitta, F.: On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var. 22(1), 289–308 (2016)

    Article  MathSciNet  Google Scholar 

  32. Oliva, F., Petitta, F.: Finite and infinite energy solutions of singular elliptic problems, existence and uniqueness. J. Differ. Equ. 264(1), 311–340 (2018)

    Article  MathSciNet  Google Scholar 

  33. Orsina, L.: Existence results for some elliptic equations with unbounded coefficients. Asymptot. Anal. 34, 187–198 (2003)

    MathSciNet  Google Scholar 

  34. Orsina, L., Petitta, F.: A Lazer–McKenna type problem with measures. Differ. Integr. Equ. 29(1–2), 19–36 (2016)

    MathSciNet  Google Scholar 

  35. Redwane, H.: Existence of solution for nonlinear elliptic equations with unbounded coefficients and \(L^1(\Omega )\) data. Int. J. Math. Math. Sci. (2009). https://doi.org/10.1155/2009/219586

  36. Sbai, A., El Hadfi, Y.: Degenerate elliptic problem with a singular nonlinearity. Complex Variables Elliptic Equ. 68, 701–718 (2021)

    Article  MathSciNet  Google Scholar 

  37. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)

    Article  MathSciNet  Google Scholar 

  38. Stuart, C.A.: Existence and approximation of solutions of non-linear elliptic equations. Math. Z. 147, 53–63 (1976)

    Article  MathSciNet  Google Scholar 

  39. Sun, Y., Zhang, D.: The role of the power 3 for elliptic equations with negative exponents. Calc. Var. Partial Differ. Equ. 49(3–4), 909–922 (2014)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to express sincere thanks to the anonymous referee for his valuable comments and suggestions that improve the manuscript.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Faculté des Sciences Juridiques, Économiques et Sociales, Université Chouaib Doukkali, El Jadida, Morocco

    Amine Marah

Authors
  1. Amine Marah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amine Marah.

Ethics declarations

Conflict of interest

The authors declare 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

Marah, A. Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term. J Elliptic Parabol Equ (2024). https://doi.org/10.1007/s41808-024-00272-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41808-024-00272-w

Keywords

Mathematics Subject Classification

Search

Navigation