Skip to main content
Log in

Gradient estimates for the p-laplace equation with a singular source

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


In this paper we will employ the method of “DeGiorgi–Nash–Moser” to establish a \(L^\infty \) local estimate for the gradient of local weak solutions to p-Laplacian equation

$$\begin{aligned} -\text{ div }\left\{ |\nabla u|^{p-2}\nabla u\right\} =f(x,u,\nabla u)\,. \end{aligned}$$

In this work we extend to \(1<p<2\) the local estimates of the \(\Vert \nabla u\Vert _\infty \) of the local weak solutions of (1.1), obtained by Bhattacharya, DiBenedetto and Manfredi in Limits as \(p\rightarrow \infty \) of \(\Delta _p u_p=f\) and related extremal problems (1989), for \(p>2\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Data availability

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


  1. In [3] the symmetric gradient \(D(\vec {u})=\frac{1}{2}\bigl (\nabla \vec {u}+(\nabla \vec {u})^t\bigr )\) appears instead of the gradient \(\nabla u\).


  1. Barrett, J.W., Liu, W.B.: Finite element approximation of the \(p\)-laplacian. Math. Comp. 61(204), 523–537 (1993)

    MathSciNet  Google Scholar 

  2. Bhattacharya, T., DiBenedetto, E., Manfredi, J.: Limits as \(p\rightarrow \infty \) of \(\Delta _p u_p=f\) and related extremal problems. Sem. Mat. Univ. Pol. Torino. Fascicolo Speciale, Nonlinear PDE’s, 15-68, (1989)

  3. Dias, G.J.: Interior estimates for power-law shear-thickening fluids. Adv. Appl. Fluid Mech. 26(1), 49–66 (2021)

    Google Scholar 

  4. DiBenedetto, E.: \(C^{1,\alpha }\)-local regularity of weak solutions of degenerate elliptic equations. Non Linear Analysis. Theory, Methods e Applications, (7), 8, 827-850, (1983)

  5. DiBenedetto, E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13(3), 487-535, (1986)

  6. DeGiorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)

    MathSciNet  Google Scholar 

  7. Dinca, G., Jebelean, P., Mawhin, J.: Variational and topological methods for dirichlet problems with \(p\)-laplacian. Port. Math. 58(3), 339–378 (2001)

    MathSciNet  Google Scholar 

  8. Duzaar, F. & Mingione, G.: Local lipschitz regularity for degenerate elliptic systems. Ann. I.l. H. Poincaré - AN 27, 1361-1396, (2010)

  9. Evans, L.C.: A new proof of local \(C^{1,\alpha }\) regularity for solutions of certain degenerate elliptic p.d.e. J. Diff. Equ. 45, 356–373 (1982)

    Article  Google Scholar 

  10. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, (1998)

  11. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations: Steady-state problems, 2nd edn. Springer-Verlag, Berlin (2011)

    Google Scholar 

  12. Ladyzenskaja, O.A., Ural’ceva, N.N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968)

    Google Scholar 

  13. Lindgren, E., Lindqvist, P.: Regularity of the \(p\)-Poisson equation in the plane. J. Anal. Math. 132, 217–228 (2017)

    Article  MathSciNet  Google Scholar 

  14. Lewis, J.L.: Regularity of the derivatives of solutions to certains degenerate elliptic equations. Indiana Univ. Math. J. 32(6), 849–858 (1982)

    Article  MathSciNet  Google Scholar 

  15. Moser, J.: A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Pure App. Math. 8, 457–468 (1960)

    Article  MathSciNet  Google Scholar 

  16. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  Google Scholar 

  17. Pimentel, E.A., Rampasso, G.C., Santos, M.S.: Improved regularity for the \(p\)-Poisson equation. Nonlinearity 33, 3050–3061 (2020)

    Article  MathSciNet  Google Scholar 

  18. Tolksdorf, P.: Everywhere regularity for some quasilinear systems with a lack of ellipticity. Ann. Mat. Pura Appl. 134, 241–266 (1983)

    Article  MathSciNet  Google Scholar 

  19. Uhlenbeck, K.: Regularity for a class of nonlinear elliptic systems. Acta Math. 138, 219–240 (1977)

    Article  MathSciNet  Google Scholar 

Download references


This work was supported by public call no 003/2017 - CAPES/FAPEAP.

Author information

Authors and Affiliations



All authors contributed to the study conception and design. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Gilberlandio J. Dias.

Ethics declarations

Financial Interests

The authors have no relevant financial or non-financial interests to disclose.

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

Dias, G.J., Duarte, Í.B.M. Gradient estimates for the p-laplace equation with a singular source. J Elliptic Parabol Equ (2024).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


Mathematics Subject Classification