Year: 2017
Author: Dongshuang Zhang
Journal of Partial Differential Equations, Vol. 30 (2017), Iss. 3 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v30.n3.3
Journal of Partial Differential Equations, Vol. 30 (2017), Iss. 3 : pp. 221–231
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 11
Keywords: Sobolev embedding