  We prove that both the liquid drop model in $\mathbb{R}^3$ with an attractive
background nucleus and the Thomas-Fermi-Dirac-von Weizs\"{a}cker (TFDW) model
attain their ground-states \emph{for all} masses as long as the external
potential $V(x)$ in these models is of long range, that is, it decays slower
than Newtonian (e.g., $V(x)\gg |x|^{-1}$ for large $|x|$.) For the TFDW model
we adapt classical concentration-compactness arguments by Lions, whereas for
the liquid drop model with background attraction we utilize a recent
compactness result for sets of finite perimeter by Frank and Lieb.
