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

Alexander Grigor’yan Department of Mathematics, University of Bielefeld, Bielefeld, 33501, Germany; School of Mathematical Sciences and Laboratory of Pure Mathematics and Combinatorics, Nankai University, Tianjin, 300071, China Yong Lin Department of Mathematics, Renmin University of China, Beijing, 100872, China YunYan Yang Department of Mathematics, Renmin University of China, Beijing, 100872, China

Analysis of PDEs mathscidoc:2207.03002

Science China Mathematics, 60, 1311–1324, 2017.4
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) + ϵg. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.
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@inproceedings{alexander2017existence,
  title={Existence of positive solutions to some nonlinear equations on locally finite graphs},
  author={Alexander Grigor’yan, Yong Lin, and YunYan Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707121610514307558},
  booktitle={Science China Mathematics},
  volume={60},
  pages={1311–1324},
  year={2017},
}
Alexander Grigor’yan, Yong Lin, and YunYan Yang. Existence of positive solutions to some nonlinear equations on locally finite graphs. 2017. Vol. 60. In Science China Mathematics. pp.1311–1324. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707121610514307558.
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