Existence and nonexistence of minimizer for Thomas-Fermi-Dirac-von Weizsäcker model on lattice graph

Abstract

The focus of our paper is to investigate the possibility of a minimizer for the Thomas-Fermi-Dirac-von Weizsäcker model on the lattice graph $Z^3$. The model is described by the following functional:$E\left(\phi \right)=\sum _{y\in Z^3}\left(\right|\nabla \phi \left(y\right){|}^2+{\left(\phi \right(y\left)\right)}^{\frac{10}{3}}-{\left(\phi \right(y\left)\right)}^{\frac{8}{3}})+\sum _{\frac{x,y\in Z^3}{y\ne x}}\frac{{\phi }^2\left(x\right){\phi }^2\left(y\right)}{|x-y|},$ with the additional constraint that $\sum _{y\in Z^3}{\phi }^2\left(y\right)=m$. We begin by establishing the existence of a minimizer for this model when m is sufficiently small. Conversely, we demonstrate that no minimizer exists when m exceeds a certain threshold. Additionally, we extend our analysis to a subset $\Omega \subset Z^3$ and prove the nonexistence of a minimizer for the following functional:$E\left(\Omega \right)=|\partial \Omega |+\sum _{\frac{x,y\in \Omega }{y\ne x}}\frac{1}{|x-y|},$ under the constraint that $\left|\Omega \right|=V$ is sufficiently large.