Sinh-Gordon equations on finite graphs

TL;DR

This paper investigates the sinh-Gordon equation on finite graphs, establishing a priori estimates, calculating topological degree, and proving the existence of solutions for nonzero prescribed functions.

Contribution

It introduces a new a priori estimate and computes the topological degree for the sinh-Gordon equation on finite graphs, leading to existence results.

Findings

Classical sinh-Gordon equation with nonzero functions is always solvable on finite graphs.

Established a uniform a priori estimate for the equation.

Calculated the topological degree case by case.

Abstract

In this paper, we focus on the sinh-Gordon equation on graphs. We introduce a uniform a priori estimate to define the topological degree for this equation with nonzero prescribed functions on finite, connected and symmetric graphs. Furthermore, we calculate this topological degree case by case and show several existence results. In particular, we prove that the classical sinh-Gordon equation with nonzero prescribed function is always solvable on such graphs.