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

TL;DR

This paper establishes existence results for solutions to the Tzitzéica and generalized Tzitzéica equations on finite graphs using topological degree theory, linking critical groups of associated functionals.

Contribution

It introduces a novel application of topological degree methods to prove existence of solutions for these equations on graphs, extending previous continuous domain results.

Findings

Existence of solutions on finite graphs proven.

Application of topological degree and critical group analysis.

Results applicable to equations with positive functions and parameters.

Abstract

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.