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The focus of our paper is to investigate the possibility of a minimizer forthe 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\neqx\hfill}\frac{{\varphi}^2(x){\varphi}^2(y)}{|x-y|}, \end{equation*} with theadditional constraint that $\sum\limits_{y\in\mathbb{Z}^{3}} {\varphi}^2(y)=m$is sufficiently small. We also prove the nonexistence of a minimizer providedthe mass $m$ is adequately large. Furthermore, we extend our analysis to asubset $\Omega \subset \mathbb{Z}^{3}$ and prove the nonexistence of aminimizer for the following functional: \begin{equation*}E(\Omega)=|\partial\Omega|+\sum_{x,y\in\Omega\atop ~y\neqx\hfill}\frac{1}{|x-y|}, \end{equation*} under the constraint that $|\Omega|=V$is sufficiently large.

Existence and nonexistence of minimizer for Thomas-Fermi-Dirac-Von Weizs\"{a}cker model on lattice graph