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, and let $\Delta _p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation \begin{equation*} -\Delta _pu+h|u|^{p-2}u=gu^{\alpha -1} \end{equation*} 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 $G$.