Volume 30, Issue 3
Semi-linear Elliptic Equations on Graph

J. Part. Diff. Eq., 30 (2017), pp. 221-231.

Published online: 2017-08

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• 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.

• Keywords

Sobolev embedding Yamabe type equation Laplacian on graph

34B45, 35A15, 58E30

@Article{JPDE-30-221, 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 = {http://global-sci.org/intro/article_detail/jpde/10466.html} }
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.