Semi-linear Elliptic Equations on Graph

Abstract

" Let G=(V,E) be a locally finite graph, Ω ⊂ V be a finite connected set, Δ be the graph Laplacian, and suppose that h : V → R is a function satisfying the coercive condition on Ω, namely there exists some constant δ > 0 such that $$∫_Ωu(-Δ+h)udμ ≥ δ ∫_Ω|∇u|²dμ,\\quad \u2200u:V → R.$$ By the mountain-pass theoremof Ambrosette-Rabinowitz, we prove that for any p > 2, there exists a positive solution to $$-Δu+hu=|u|^{p-2}u\\quad\\;\\; in\\;\\; Ω$$. Using the same method, we prove similar results for the p-Laplacian equations. This partly improves recent results of Grigor'yan-Lin-Yang."