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.
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机译：设g =（v，e）是局部有限的图形，ωcv是有限的连接集，△是图表拉普拉斯，并假设h：v→ir是满足Ω上的幂条件的函数，即存在一些恒定Δ> 0使得∫ωu（△+ h）UDμ≥Δ∫ω|2Dμ，∨U：V→Ir.By of Ambrosette-Rabinowitz的山景定理，我们证明了任何P> 2 ，存在阳性解决方案 - △U+ hu = | U | p-2u在ω.using相同的方法中，我们证明了P-Laplacian方程的类似结果。本部分提高了Grigor'yan-Lin的最近结果 - 杨。
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