@Article{JPDE-30-3,
author = {Zhang, Dongshuang},
title = {Semi-linear Elliptic Equations on Graph},
journal = {Journal of Partial Differential Equations},
year = {2017},
volume = {30},
number = {3},
pages = {221--231},
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 $$∫_Ωu(-Δ+h)udμ ≥ δ ∫_Ω|∇u|²dμ,\quad ∀u:V → R.$$ 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.},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v30.n3.3},
url = {https://global-sci.com/article/88232/semi-linear-elliptic-equations-on-graph}
}