Yamabe type equations on graphs

Abstract

Let $G=(V,E)$ be a locally finite graph, $\Omega \subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and ${\lambda }_1(\Omega )$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $\alpha <{\lambda }_1(\Omega )$, then for any $p>2$, there exists a positive solution to $-\Delta u-\alpha u=|u{|}^{p-2}u$ in ${\Omega }^{\circ }$, $u=0$ on $\partial \Omega$, where ${\Omega }^{\circ }$ and $\partial \Omega$ denote the interior and the boundary of $\Omega$ respectively. Also we consider similar problems involving the $p$-Laplacian and poly-Laplacian by the same method. Such problems can be viewed as discrete versions of the Yamabe type equations on Euclidean space or compact Riemannian manifolds.

17 pages