Existence results for Tzitz\'eica equation via topological degree method on graphs

We derive some existence results for the solutions of the Tzitz\'eica equation \begin{equation*} -\Delta u + h_1(x)e^{Au} + h_2(x)e^{-Bu}=0 \end{equation*} and the generalized Tzitz\'eica equation \begin{equation*} -\Delta u + h_1(x)e^{Au}(e^{Au}-1)+h_2(x)e^{-Bu}(e^{-Bu}-1)=0 \end{equation*} on any connected finite graph $G=(V, E)$. Here, $h_1(x)>0$, $h_2(x)>0$ are two given functions on $V$, and $A, B>0$ are two constants. Our approach involves computing the topological degree and using the connection between the degree and the critical group of an associated functional.