  We establish Zaremba problem for Laplacian and $p$-Laplacian with degenerate
weights when the Dirichlet condition is only imposed in a set of positive
weighted capacity. We prove weighted Sobolev-Poincar\'{e} inequality with sharp
scaling-invariant constants involving weighted capacity. Then we show higher
integrability of the gradient of the solution (Meyers estimate) with minimal
conditions on the part of the boundary where the Dirichlet condition is
assumed. Our results are new both for the linear $p=2$ and nonlinear case and
include problems with the weight not only as a measure but also as a multiplier
of the gradient of the solution.
