Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 18, pp. 1-20.

Title: Existence of a minimizer for the quasi-relativistic Kohn-Sham model

Authors: Carlos Argaez    (Dublin Institute of Technology, Ireland)
         Michael Melgaard (Dublin Institute of Technology, Ireland)

Abstract: 
 We study the standard and extended Kohn-Sham models for
 quasi-relativistic N-electron Coulomb systems; that is,
 systems where the kinetic energy of the electrons is given
 by the quasi-relativistic operator
 $$
 \sqrt{-\alpha^{-2}\Delta_{x_n}+\alpha^{-4}}-\alpha^{-2}.
 $$
 For spin-unpolarized systems in the local density approximation,
 we prove existence of a ground state (or minimizer) provided that
 the total charge $Z_{\rm tot}$ of K nuclei is greater than
 N-1 and that $Z_{\rm tot}$ is smaller than a critical
 charge $Z_{\rm c}=2 \alpha^{-1} \pi^{-1}$.

Submitted April 12, 2011. Published January 30, 2012.
Math Subject Classifications: 35J60, 47J10, 58Z05, 81V55.
Key Words: Kohn-Sham equations; ground state;  variational methods;
           concentration-compactness; density operators.