Yamabe type equations on finite graphs

Let G = (V, E) be a locally finite graph, Ω ⊂ V be a bounded open domain, ∆ be the usual graph Laplacian, and λ1(Ω) be the first eigenvalue of −∆ with respect to Dirichlet boundary condition. Using the mountain pass theorem of Ambrosette and Rabinowitz, We prove that if α < λ1(Ω), then for any p > 2, there exists a positive solution to  −∆u − αu = |u| p−2u in Ω◦, u = 0 on ∂Ω, where Ω◦ and ∂Ω denote the interior and the boundary of Ω 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 equation on Euclidean space or compact Riemannian manifolds.