Abstract
Let \(G=(V,E)\) be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on G (the Schrödinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.
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Acknowledgements
We thank the reviewers for their careful reading and valuable comments. Yong Lin is supported by the NSFC (Grant No. 12071245). Yunyan Yang is supported by the NSFC (Grant No. 11721101) and the National Key Research and Development Project SQ2020YFA070080. Both of the two authors are supported by the NSFC (Grant No. 11761131002).
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Lin, Y., Yang, Y. Calculus of variations on locally finite graphs. Rev Mat Complut 35, 791–813 (2022). https://doi.org/10.1007/s13163-021-00405-y
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DOI: https://doi.org/10.1007/s13163-021-00405-y