Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

We study regularity for solutions of quasilinear elliptic equations of the form $÷\text{A}(x,u,\mathrm{\nabla }u)=÷\text{F}$ in bounded domains in ${\mathbb{R}}^n$. The vector field $\text{A}$ is assumed to be continuous in $u$, and its growth in $\nabla u$islikethatofthe$p$-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 $\text{A}$. As a consequence, we obtain that $\mathrm{\nabla }u$ is in the classical Morrey space ${\text{calM}}^{q,\lambda }$ or weighted space $L_w^q$ whenever $\mathrm{\mid }\text{F}{\mathrm{\mid }}^{\frac{1}{p-1}}$ is respectively in ${\text{calM}}^{q,\lambda }$ or $L_w^q$, 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 $\mathrm{\nabla }u$ in a weighted Morrey space that is different from the functional space that the data $\mathrm{\mid }\text{F}{\mathrm{\mid }}^{\frac{1}{p-1}}$ belongs to.