Sobolev estimates for singular-degenerate quasilinear equations beyond the $A_2$ class

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $\mathrm{\Omega }$ whose coefficients can be degenerate or singular of the type $\text{dist}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{\alpha }$, where $\mathrm{\partial }\mathrm{\Omega }$ is the boundary of $\mathrm{\Omega }$ and $\alpha \in (-1,\mathrm{\infty })$ is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.