  This paper studies the Sobolev regularity estimates for weak solutions of a
class of degenerate, and singular quasi-linear elliptic problems of the form
$\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with
non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains.
The coefficients $\mathbf{A}$ could be be singular, and degenerate or both in
$x$ in the sense that they behave like some weight function $\mu$, which is in
the $A_2$ class of Muckenhoupt weights. Global and interior weighted
$W^{1,p}(\Omega, \omega)$-regularity estimates are established for weak
solutions of these equations with some other weight function $\omega$. The
results obtained are even new for the case $\mu =1$ because of the dependence
on the solution $u$ of $\mathbf{A}$. In case of linear equations, our
$W^{1,p}$-regularity estimates can be viewed as the Sobolev's counterpart of
the H\"{o}lder's regularity estimates established by B. Fabes, C. E. Kenig, and
R. P. Serapioni.
