Dirichlet problem for plurisubharmonic functions on bounded quasi B-regular domains in $\mathbb{C}^n$

We investigate existence and uniqueness of maximal plurisubharmonic functions on bounded domains with boundary data that are not assumed to be continuous or bounded. The result is applied to approximate (possibly unbounded from above) plurisubharmonic functions by continuous quasi upper bounded ones. A key step in our approach is to explore continuity of the Perron-Bremermann envelope of plurisubharmonic functions that are dominated by a given function $\phi$ defined on the closure of the domain.