{"paper":{"title":"On finite-energy solutions of Kazan-Warner equations on the lattice graph","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bobo Hua, Huyuan Chen","submitted_at":"2025-09-01T06:24:11Z","abstract_excerpt":"We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - \\Delta u= \\varepsilon e^{\\kappa u} +\\beta\\delta_0\\quad {\\rm in}\\ \\mathbb{Z}^2,$$ where $\\varepsilon=\\pm1$, $\\kappa>0$ and $\\beta\\in\\mathbb{R}$.\n  When $\\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $\\kappa$. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation.\n  When $\\varepsilon=-1$ and $\\beta>\\frac{4\\pi}{\\kappa}$, we prove that the set of finite-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01155","kind":"arxiv","version":2},"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/2509.01155/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"}