Skip to main content
SpringerLink
Log in

Nonlinear weighted elliptic equations with Sobolev weights

  • Published:
Bollettino dell'Unione Matematica Italiana Aims and scope Submit manuscript

Abstract

In this paper we are going to prove existence of solutions in \(W_{0}^{1,p}(\Omega )\) for the weighted elliptic equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathrm{div}(s(x)\, |\nabla u|^{p-2}\,\nabla u) = f(x) &{}\quad \text{ in } \Omega ,\\ u = 0 &{}\quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

under the assumption that the weight function s(x) is strictly positive and belongs to \(W^{1,p}(\Omega )\). The proof will be based on a new technique, which takes advantage of the fact that s(x) is in a Sobolev space, and that will also be applied to other equations.

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.

Similar content being viewed by others

References

  1. Arcoya, D., Boccardo, L.: Regularizing effect of the interplay between coefficients in some elliptic equations. J. Funct. Anal. 268, 1153–1166 (2015)

    Article  MathSciNet  Google Scholar 

  2. Boccardo, L.: The Fatou lemma approach to the existence in quasilinear elliptic equations with natural growth terms. Complex Var. Ellipt. Equ. 55, 445–453 (2010)

    Article  MathSciNet  Google Scholar 

  3. Boccardo, L., Cirmi, G.R.: Some existence and regularity results by Neil Trudinger revisited without the weighted Sobolev spaces framework. Bull. Inst. Math. Acad. Sin. (N.S.) 9, 295–315 (2014)

  4. Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19, 581–597 (1992)

    Article  MathSciNet  Google Scholar 

  5. Boccardo, L., Murat, F., Puel, J.-P.: Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugal Math. 41(1982), 507–534 (1984)

    MATH  Google Scholar 

  6. Boccardo, L., Orsina, L.: Strong maximum principle for some quasilinear Dirichlet problems having natural growth terms. Adv. Nonlinear Stud. 20, 503–510 (2020)

  7. Brezis, H., Ponce, A.C.: Remarks on the strong maximum principle. Differ. Integr. Equ. 16, 1–12 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Buccheri, S.: Some nonlinear elliptic problems with \(L^1(\Omega )\) coefficients. Nonlinear Anal. 177, 135–152 (2018)

    Article  MathSciNet  Google Scholar 

  9. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Dover Publications, xii+404 (2006)

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

    Article  Google Scholar 

  11. Trudinger, N.S.: Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients. Math. Z. 156, 291–301 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sapienza Università di Roma-Istituto Lombardo, P.le A. Moro 2, Rome, Italy

    Lucio Boccardo

  2. Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, Rome, Italy

    Pasquale Imparato & Luigi Orsina

Authors
  1. Lucio Boccardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pasquale Imparato
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luigi Orsina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luigi Orsina.

Ethics declarations

Conflict of interest

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

Boccardo, L., Imparato, P. & Orsina, L. Nonlinear weighted elliptic equations with Sobolev weights. Boll Unione Mat Ital 15, 503–514 (2022). https://doi.org/10.1007/s40574-021-00314-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40574-021-00314-4

Search

Navigation