Continuity of envelopes of unbounded plurisubharmonic functions

On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $\varphi$ such that ${\varphi }^{\ast }={\varphi }_{\ast }$ on the closure of the domain. For $\varphi$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P\varphi$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge–Amp`ere equation $$(d{d}^{c}u{)}^n=\mu$$ being uniquely solvable for continuous boundary data implies that it is also uniquely solvable for a class of boundary values unbounded from above.