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 and 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.