We study quasilinear elliptic equations of the form $\text{ div} \mathbf{ A}(x,u,\nabla u) = \text{ div}\mathbf{ F} $ in bounded domains in $\mathbb{ R}^n$, $n\geq 1$. The vector field $\mathbf{ A}$ is allowed to be discontinuous in $x$, Lipschitz continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1<p<\infty$. We establish interior $W^{ 1,q}$-estimates for locally bounded weak solutions to the equations for every $q>p$, and we show that similar results also hold true in the setting of { \it Orlicz} spaces. Our regularity estimates extend results which are only known for the case $\mathbf{ A}$ is independent of $u$ and they complement the well-known interior $C^{ 1,\alpha}$- estimates obtained by DiBenedetto \cite{ D} and Tolksdorf \cite{ T} for general quasilinear elliptic equations.
Categories: math.AP