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 \(\Omega \) $\mathrm{\Omega }$ whose coefficients can be degenerate or singular of the type \(\text {dist}(x, \partial \Omega )^\alpha \) $\text{dist}{(x,\partial \mathrm{\Omega })}^{\alpha }$ , where \(\partial \Omega \) $\partial \mathrm{\Omega }$ is the boundary of \(\Omega \) $\mathrm{\Omega }$ and \(\alpha \in (-1, \infty )\) $\alpha \in (-1,\infty )$ 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.