{"paper":{"title":"Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Daniele Bartolucci","submitted_at":"2012-08-26T15:22:54Z","abstract_excerpt":"We consider the mean field equation: (1) \\Delta u+\\rho\\frac{e^u}{\\int_\\Omega e^u}=0 & \\hbox{in} \\;\\Omega, u=0 & \\hbox{on}\\;\\partial\\Omega, where $\\Omega\\subset \\mathbb{R}^2$ is an open and bounded domain of class $C^1$. In his 1992 paper, Suzuki proved that if $\\Omega$ is a simply-connected domain, then equation (1) admits a unique solution for $\\rho\\in[0,8\\pi)$. This result for $\\Omega$ a simply-connected domain has been extended to the case $\\rho=8\\pi$ by Chang, Chen and the second author. However, the uniqueness result for $\\Omega$ a multiply-connected domain has remained a long standing op"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5228","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"}