{"paper":{"title":"Existence of solutions to a generalized self-dual Chern-Simons equation on graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Peirong Zhong, Yingshu L\\\"u","submitted_at":"2021-07-27T00:57:38Z","abstract_excerpt":"Let $ G=(V,E) $ be a connected finite graph and $ \\Delta $ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \\begin{eqnarray}\\label{one1} \\Delta{u}=-\\lambda{e^{F(u)}[e^{F(u)}-1]^2}+4\\pi\\sum_{i=1}^{M}{\\delta_{p_{j}}}, \\end{eqnarray} where \\begin{equation} F(u)=\\left\\{\\begin{array}{l}\n  \\widetilde{F}(u), \\ \\quad u\\leq0,\n  0, \\quad \\quad \\quad u>0,\n  \\end{array} \\right. \\end{equation} $ \\widetilde{F}(u) $ satisfies $ u=1+{\\widetilde {F}(u)}-e^{\\widetilde {F}(u)} $, $ \\lambda>0 $, $M$ is any fixed positive integer, $ \\delta_{p_{j}}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.12535","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2107.12535/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}