  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.
