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 $\overline{\Omega}$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $H^2(\Omega ,\varphi)$ with a plurisubharmonic weight $\varphi$. As an application, we show that the Hartogs domain $$\Omega _\alpha : = \{(z,w) \in D\times \C: |w|< \delta^\alpha_D(z) \}, \ \ \ \alpha>0,$$ where $D\subset \subset \C$ and $\delta_D$ denotes the boundary distance, is Bergman complete if and only if every boundary point of $D$ is non-isolated.