Existence domains for holomorphic $L^p$-functions

If $\Omega$ is a domain of holomorphy in $\Bbb C^n$, having a compact topological closure into another domain of holomorphy $U\subset \Bbb C^n$ such that $(\Omega,U)$ is a Runge pair, we construct a function $F$ holomorphic in $\Omega$ which is singular at every boundary point of $\Omega$ and such that $F$ is in $L^p(\Omega)$, for any $p\in (0,+\infty)$.