Yamabe type equations on graphs
  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. 
Alexander Grigor'yan, Yong Lin, Yunyan Yang
math.AP