*preprint*

**Inserted:** 5 oct 2021

**Year:** 2018

**Abstract:**

We study regularity for solutions of quasilinear elliptic equations of the
form $\div \A(x,u,\nabla u) = \div \F $ in bounded domains in $\R^n$. The
vector field $\A$ 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 $\A$. As a consequence, we obtain that
$\nabla u$ is in the classical Morrey space $\calM^{q,\lambda}$ or weighted
space $L^q_w$ whenever $

\F

^{\frac{1}{p-1}}$ is respectively in
$\calM^{q,\lambda}$ or $L^q_w$, where $q$ is any number greater than $p$ and
$w$ is any weight in the Muckenhoupt class $A_{\frac{q}{p}}$. 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 $

\F

^{\frac{1}{p-1}}$ belongs to.