Density in weighted Bergman spaces and Bergman completeness of Hartogs domains

We study the density of functions which are holomorphic in a neighbourhood of the closure $\bar{\mathrm{\Omega }}$ of a bounded non-smooth pseudoconvex domain $\mathrm{\Omega }$, in the Bergman space $H^2(\mathrm{\Omega },\phi )$ with a plurisubharmonic weight $\phi$. As an application, we show that the Hartogs domain $${\mathrm{\Omega }}_{\alpha }:=\{(z,w)\in D\times \text{C}:\mathrm{\mid }w\mathrm{\mid }<{\delta }_D^{\alpha }(z)\},\alpha >0,$$ where $D\subset \subset \text{C}$ and ${\delta }_D$ denotes the boundary distance, is Bergman complete if and only if every boundary point of $D$ is non-isolated.