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