Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

Abstract

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