{"paper":{"title":"Yamabe type equations on graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander Grigor'yan, Yong Lin, Yunyan Yang","submitted_at":"2016-07-15T14:20:39Z","abstract_excerpt":"Let $G=(V,E)$ be a locally finite graph, $\\Omega\\subset V$ be a bounded domain, $\\Delta$ be the usual graph Laplacian, and $\\lambda_1(\\Omega)$ be the first eigenvalue of $-\\Delta$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $\\alpha<\\lambda_1(\\Omega)$, then for any $p>2$, there exists a positive solution to $-\\Delta u-\\alpha u=|u|^{p-2}u$ in $\\Omega^\\circ$, $u=0$ on $\\partial\\Omega$, where $\\Omega^\\circ$ and $\\partial\\Omega$ denote the interior and the boundary of $\\Omega$ respectively. Also we consider similar pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04521","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":""},"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"}