We prove that both the liquid drop model in R^3 with an attractive background nucleus and the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) model attain their ground-states 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)≫ |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

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... mpactness result for sets of finite perimeter by Frank and Lieb.