Resolutivity and invariance for the Perron method for degenerate equations of divergence type

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation $\mathcal{A}(x,\mathrm{\nabla }u)=0$ in a bounded open set $\mathrm{\Omega }\subset {\mathbf{R}}^n$. The vector-valued function $\mathcal{A}$ satisfies the standard ellipticity assumptions with a parameter $1<p<\mathrm{\infty }$ 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.