Positive Solutions of p-th Yamabe Type Equations on Infinite Graphs

TL;DR

This paper proves the existence of at least one positive solution for a class of nonlinear p-th Yamabe type equations on infinite, locally finite weighted graphs, extending concepts from smooth manifold equations to discrete graph settings.

Contribution

It establishes the existence of positive solutions for p-th Yamabe type equations on infinite graphs, a novel extension from continuous to discrete geometric analysis.

Findings

Existence of positive solutions on infinite graphs

Extension of Yamabe equation concepts to discrete settings

Application of variational methods on graphs

Abstract

Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}$$ on $G$, where $h$ and $g$ are known, $2<\alpha\leq p$. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on $G$.