TY - JOUR
T1 - Semi-linear Elliptic Equations on Graph
AU - Zhang , Dongshuang
JO - Journal of Partial Differential Equations
VL - 3
SP - 221
EP - 231
PY - 2017
DA - 2017/08
SN - 30
DO - http://doi.org/10.4208/jpde.v30.n3.3
UR - https://global-sci.org/intro/article_detail/jpde/10466.html
KW - Sobolev embedding
KW - Yamabe type equation
KW - Laplacian on graph
AB -  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 $$∫_Ωu(-Δ+h)udμ ≥ δ ∫_Ω|∇u|²dμ,\quad ∀u:V → R.$$ 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.