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

Abstract

We prove that both the liquid drop model in ${\mathbb{R}}^3$ with an attractive background nucleus and the Thomas-Fermi-Dirac-von Weizsäcker (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.

Added minor corrections, and simplified proof of Theorem 2. This version is to appear in the Journal of Mathematical Physics

References in corpus (2)

• MOND theory
• Mond - a review