  The focus of our paper is to investigate the possibility of a minimizer for
the Thomas-Fermi-Dirac-von Weizs\"{a}cker model on the lattice graph
$\mathbb{Z}^{3}$. The model is described by the following functional:
\begin{equation*}
E(\varphi)=\sum_{y\in\mathbb{Z}^{3}}\left(|\nabla\varphi(y)|^2+
(\varphi(y))^{\frac{10}{3}}-(\varphi(y))^{\frac{8}{3}}\right)+
\sum_{x,y\in\mathbb{Z}^{3}\atop ~\ y\neq
x\hfill}\frac{{\varphi}^2(x){\varphi}^2(y)}{|x-y|}, \end{equation*} with the
additional constraint that $\sum\limits_{y\in\mathbb{Z}^{3}} {\varphi}^2(y)=m$
is sufficiently small. We also prove the nonexistence of a minimizer provided
the mass $m$ is adequately large. Furthermore, we extend our analysis to a
subset $\Omega \subset \mathbb{Z}^{3}$ and prove the nonexistence of a
minimizer for the following functional: \begin{equation*}
E(\Omega)=|\partial\Omega|+\sum_{x,y\in\Omega\atop ~y\neq
x\hfill}\frac{1}{|x-y|}, \end{equation*} under the constraint that $|\Omega|=V$
is sufficiently large.
