Positive Solutions of p-th Yamabe Type Equations on Infinite 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 $$-{\Delta }_{p}u+h|u{|}^{p-2}u=g{u}^{\alpha -1}$$ on $G$, where $h$ and $g$ are known, $2<\alpha \le p$. 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 $G$.