  We consider Perron solutions to the Dirichlet problem for the quasilinear
elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u) = 0$ in a bounded
open set $\Omega\subset\mathbf{R}^n$. The vector-valued function $\mathcal{A}$
satisfies the standard ellipticity assumptions with a parameter $1<p<\infty$
and a $p$-admissible weight $w$. We show that arbitrary perturbations on sets
of $(p,w)$-capacity zero of continuous (and certain quasicontinuous) boundary
data $f$ are resolutive and that the Perron solutions for $f$ and such
perturbations coincide. As a consequence, we prove that the Perron solution
with continuous boundary data is the unique bounded solution that takes the
required boundary data outside a set of $(p,w)$-capacity zero.
