The Einstein-scalar field Lichnerowicz equations on graphs

In this article, we consider the Einstein-scalar field Lichnerowicz equation \(\begin{aligned} -\Delta u+hu=Bu^{p-1}+Au^{-p-1} \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+hu=B{u}^{p-1}+A{u}^{-p-1}\end{array}$$ on any connected finite graph \(G=(V,E)\) $G=(V,E)$ , where A, B, h are given functions on V with \(A\ge 0\) $A\ge 0$ , \(A\not \equiv 0\) $A\not\equiv 0$ on V, and \(p>2\) $p>2$ is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely \(h>0\) $h>0$ , \(h<0\) $h<0$ and \(h=0\) $h=0$ respectively.