Semi-linear Elliptic Equations on Graph

摘要

Let G =(V,E) be a locally finite graph,Ω C V be a finite connected set,△ be the graph Laplacian,and suppose that h:V → IR is a function satisfying the coercive condition on Ω,namely there exists some constant δ ＞ 0 such that ∫Ωu(△+h)udμ≥δ∫Ω |▽u|2dμ,∨u:V→IR.By the mountain-pass theorem of Ambrosette-Rabinowitz,we prove that for any p ＞ 2,there exists a positive solution to -△u+hu=|u|p-2u 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.