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.
Categories: math.AP