In this talk, we will discuss weighted $L^q$, $1<q<2$, estimates for nonlinear elliptic equations,

as an application of $L^p$ estimates, $1<p<\infty$, for linear ellitpic equations.

We will assume that the nonlinearity $A(x, \xi)$ satisfies linear growth condtions

and that $A(x, \xi)$ is asymptotically Uhlenbeck, i.e., $|A(x, \xi) - A(x) \xi| \leq \epsilon |\xi|$

for sufficiently large $|\xi|$.