Existence of positive solutions to some nonlinear equations on locally finite graphs |
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Authors: | Alexander Grigor’yan Yong Lin YunYan Yang |
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Institution: | 1.Department of Mathematics,University of Bielefeld,Bielefeld,Germany;2.School of Mathematical Sciences and Laboratory of Pure Mathematics and Combinatorics,Nankai University,Tianjin,China;3.Department of Mathematics,Renmin University of China,Beijing,China |
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Abstract: | Let G = (V, E) be a locally finite graph, whose measure μ(x) has positive lower bound, and Δ 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 Δu + 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 Δu + hu = f(x, u) + ?g. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds. |
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