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) + ϵg. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.
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Acknowledgements
This work was supported by Sondernforschungsbereich (SFB) 701 of the German Research Council and National Natural Science Foundation of China (Grant Nos. 11271011 and 11171347). The authors thank the referees for their careful reading and constructive comments on this paper.
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Grigor’yan, A., Lin, Y. & Yang, Y. Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci. China Math. 60, 1311–1324 (2017). https://doi.org/10.1007/s11425-016-0422-y
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DOI: https://doi.org/10.1007/s11425-016-0422-y