Yamabe type equations on graphs

Let G=(V,E)G=(V,E) be a locally finite graph, Ω⊂VΩ⊂V be a bounded domain, Δ be the usual graph Laplacian, and λ1(Ω)λ1(Ω) be the first eigenvalue of −Δ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti–Rabinowitz, we prove that if α<λ1(Ω)α<λ1(Ω), then for any p>2p>2, there exists a positive solution to{−Δu−αu=|u|p−2uinΩ∘,u=0on∂Ω, where Ω∘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 equations on Euclidean space or compact Riemannian manifolds.