The $1$-Yamabe equation on graph

TL;DR

This paper investigates the $1$-Yamabe equation on finite graphs involving the discrete $1$-Laplacian and demonstrates the existence of nontrivial solutions under certain conditions.

Contribution

The paper establishes the existence of nontrivial solutions to the $1$-Yamabe equation on finite graphs, extending the understanding of nonlinear equations involving the discrete $1$-Laplacian.

Findings

Existence of nontrivial solutions $u eq0$ with $u extgreater 0$

Solutions are guaranteed for the given $1$-Yamabe equation on finite graphs

The equation always admits a nontrivial solution under the specified conditions

Abstract

We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above $1$-Yamabe equation always has a nontrivial solution $u\geq0$, $u\neq0$.