A p-th Yamabe equations on graph

TL;DR

This paper proves the existence of positive solutions for a class of p-th Yamabe equations on finite graphs, extending the understanding of nonlinear equations involving the p-Laplacian in discrete settings.

Contribution

It establishes the existence of solutions for a p-th Yamabe equation on finite graphs, a novel result in the discrete nonlinear analysis field.

Findings

Positive solutions exist for the p-th Yamabe equation on finite graphs.

The solutions are guaranteed for some real parameter λ.

The results extend continuous Yamabe problem insights to discrete graph settings.

Abstract

Assume $\alpha\geq p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: $$\Delta_p\varphi+h\varphi^{p-1}=\lambda f\varphi^{\alpha-1},$$ where $\Delta_p$ is the discrete $p$-Laplacian, $h$ and $f>0$ are fixed real functions defined on all vertices. We show that the above equation always has a positive solution $\varphi$ for some constant $\lambda\in\mathds{R}$.