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arXiv · 2505.19704

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

Kaizhe Chen·Heng Zhang
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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.

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Kaizhe Chen, Heng Zhang. 2025-05-26. Existence results for Tzitz\'eica equation via topological degree method on graphs. https://arxiv.org/abs/2505.19704

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