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

We consider Perron solutions to the Dirichlet problem for quasilinear elliptic equation divA(x,∇u)=0 in a bounded open set Ω⊂Rn. The vector-valued function A satisfies standard ellipticity assumptions with 1<p<∞ and p-admissible weight w. show that arbitrary perturbations on sets of (p,w)-capacity zero continuous (and certain quasicontinuous) boundary data f are resolutive such coincide. As consequence, we prove solution is unique takes required outside zero.