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

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 ${\mathbb{Z}}^3$. 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 $\sum _{y\in {\mathbb{Z}}^3}{\phi }^2(y)=m$ 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 $\mathrm{\Omega }\subset {\mathbb{Z}}^3$ and prove the nonexistence of a minimizer for the following functional: \begin{equation*} E(Ω)=|\partialΩ|+\sum_{x,y\inΩ\atop ~y\neq x\hfill}\frac{1}{|x-y|}, \end{equation*} under the constraint that $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=V$ is sufficiently large.