Mass splitting in the Thomas–Fermi–Dirac–von Weizsäcker model with background potential

We consider minimization problems of the Thomas–Fermi–Dirac–von Weizsäcker (TFDW) type in which the Newtonian potential is perturbed by a background potential satisfying mild conditions, which ensures the existence of minimizers. We describe the structure of minimizing sequences for those variants and obtain a more precise characterization of patterns in minimizing sequences for the TFDW functionals regularized by long-range perturbations.