JournalsrmiVol. 36, No. 6pp. 1627–1658

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

  • Giuseppe Di Fazio

    Università degli Studi di Catania, Italy
  • Truyen Nguyen

    University of Akron, USA
Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations cover

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Abstract

We study regularity for solutions of quasilinear elliptic equations of the form divA(x,u,u)=divF\mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F} in bounded domains in Rn\mathbb{R}^n. The vector field A\mathbf{A} is assumed to be continuous in uu, and its growth in u\nabla u is like that of the pp-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions uu to the equation under a small BMO condition in xx for A\mathbf{A}. As a consequence, we obtain that u\nabla u is in the classical Morrey space Mq,λ\mathcal{M}^{q,\lambda} or weighted space LwqL^q_w whenever F1/(p1)|\mathbf{F}|^{1/(p-1)} is respectively in Mq,λ\mathcal{M}^{q,\lambda} or LwqL^q_w, where qq is any number greater than pp and ww is any weight in the Muckenhoupt class Aq/pA_{q/p}. In addition, our two-weight estimate allows the possibility to acquire the regularity for u\nabla u in a weighted Morrey space that is different from the functional space that the data F1/(p1)|\mathbf{F}|^{1/(p-1)} belongs to.

Cite this article

Giuseppe Di Fazio, Truyen Nguyen, Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations. Rev. Mat. Iberoam. 36 (2020), no. 6, pp. 1627–1658

DOI 10.4171/RMI/1178