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Existence of solutions to the nonlinear Schrödinger equation on locally finite graphs

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Abstract

Let \(G=(V, E)\) be a locally finite connected graph and \(\Delta \) be the usual graph Laplacian operator. According to Lin and Yang (Rev. Mat. Complut., 2022), using calculus of variations from local to global, we establish the existence of solutions to the nonlinear Schrödinger equation on locally finite graphs, say \(-\Delta u+hu=fe^u\), \(x\in V\). In particular, we suppose that there exist positive constants \(\mu _0\) and \(\omega _{0}\) such that the measure \(\mu (x)\ge \mu _0\) for \(x\in V\) and symmetric weight \(\omega _{xy}\ge \omega _0\) for all \(xy\in E\), if h and f satisfy distinct certain assumptions, we prove that the above-mentioned equation has a strictly negative solution by three cases.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Zidong Qiu & Yang Liu

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  1. Zidong Qiu
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  2. Yang Liu
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Correspondence to Yang Liu.

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Qiu, Z., Liu, Y. Existence of solutions to the nonlinear Schrödinger equation on locally finite graphs. Arch. Math. 120, 403–416 (2023). https://doi.org/10.1007/s00013-023-01830-9

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