Ground-states for the liquid drop and TFDW models with long-range attraction

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., $V(x)\gg \mathrm{\mid }x{\mathrm{\mid }}^{-1}$ for large $\mathrm{\mid }x\mathrm{\mid }$.) 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.