Existence results for Tzitzéica equation via topological degree method on graphs

TL;DR

This work investigates existence (and multiplicity) of solutions to the Tzitzéica equation on connected finite graphs and its generalized form with positive weight functions. It leverages topological degree theory in the discrete setting, constructing deformation arguments to compute Brouwer degrees and relating them to critical groups of an associated energy functional. The authors obtain explicit degree values: $\mathbf{d}_{h_1,h_2}=1$ when $h_2<0$ (with $h_1>0$) and $\mathbf{d}_{h_1,h_2}=0$ when $h_2>0$ (with $h_1>0$); and show $\mathbf{\tilde{d}}_{h_1,h_2}=0$ for the generalized equation. They further prove existence of at least one solution under certain inequalities between $A,B$ and $h_1,h_2$, and, via variational methods and critical groups, establish the existence of at least two solutions in those regimes. These results extend nonlinear elliptic PDE existence and multiplicity theory to finite graphs, highlighting the power of degree theory and variational methods on discrete structures.

Abstract

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.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4