Existence results for Tzitzéica equation via topological degree method on graphs

We derive some existence results for the solutions of the Tzitzéica equation \begin{equation*} -Δu + h_1(x)e^{Au} + h_2(x)e^{-Bu}=0 \end{equation*} and the generalized Tzitzéica equation \begin{equation*} -Δu + 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.