Semi-linear Elliptic Equations on Graph

Semi-linear Elliptic Equations on Graph

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.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

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

Author Details

Dongshuang Zhang

  1. Existence results for some nonlinear elliptic systems on graphs

    Man, Shoudong

    Journal of Mathematical Analysis and Applications, Vol. 543 (2025), Iss. 2 P.128973

    https://doi.org/10.1016/j.jmaa.2024.128973 [Citations: 0]
  2. Existence and Multiplicity of Nontrivial Solutions for a $(p,q)$-Laplacian System on Locally Finite Graphs

    Yang, Ping | Zhang, Xingyong

    Taiwanese Journal of Mathematics, Vol. 28 (2024), Iss. 3

    https://doi.org/10.11650/tjm/240201 [Citations: 0]
  3. Existence and uniqueness theorems for some semi-linear equations on locally finite graphs

    Pinamonti, Andrea | Stefani, Giorgio

    Proceedings of the American Mathematical Society, Vol. 150 (2022), Iss. 11 P.4757

    https://doi.org/10.1090/proc/16046 [Citations: 6]