Existence of positive solutions to some nonlinear equations on locally finite graphs

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2017,（7）
• 页码范围: 1311-1324
• 页数: 14页
• 语种: 英文
• DOI: 10.1007/s11425-016-0422-y

摘要

Let G =(V,E) be a locally finite graph,whose measure μ(x) has positive lower bound,and △ be the usual graph Laplacian.Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz (1973),we establish existence results for some nonlinear equations,namely △u + hu =f(x,u),x ∈ V.In particular,we prove that if h and f satisfy certain assumptions,then the above-mentioned equation has strictly positive solutions.Also,we consider existence of positive solutions of the perturbed equation △u + hu =f(x,u) + eg.Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.