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.
Categories: math.AP