The Perron--Wiener--Brelot method is a powerful way of solving the
 Dirichlet problem in general domains. For every boundary data it
 provides an upper and a lower solution. In the case of the Laplace
 equation it is well known when these two solutions coincide and
 resolutive boundary data are characterized by means of the harmonic
 measure. For the nonlinear $p$-Laplace equation, the class of resolutive
 boundary data is not known, but there are some partial results, e.g.
 that continuous boundary data are resolutive. The invariance under
 perturbations on sets of capacity zero is also not fully understood in
 the nonlinear case. In the talk we shall see how some of these questions
 can be attacked by changing the metric in the domain and by considering
 new capacities.