Yamabe type equations on graphs

Let G = (V, E) be a locally finite graph, Omega subset of V be a bounded domain, A 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-au = vertical bar u vertical bar(p-2)u in Omega degrees u = 0 on partial derivative Omega, where Omega degrees and partial derivative 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. (C) 2016 Elsevier Inc. All rights reserved.