{"paper":{"title":"On the supercritical mean field equation on pierced domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, Mohameden Ould Ahmedou","submitted_at":"2013-12-13T10:50:59Z","abstract_excerpt":"We consider the problem $$(P_\\eps)\\qquad\n  \\Delta u +\\lambda {e^{u}\\over\\int\\limits_{\\Omega\\setminus B(\\xi,\\eps)}e^u}=0\\ \\hbox{in}\\ \\Omega\\setminus B(\\xi,\\eps),\\quad u =0\\ \\hbox{on}\\ \\partial\\(\\Omega\\setminus B(\\xi,\\eps)\\), $$ where $\\Omega$ is a smooth bounded open domain in $\\rr^2$ which contains the point $\\xi.$ We prove that if $\\lambda>8\\pi,$ problem $(P_\\eps)$ has a solutions $u_\\eps$ such that $$u_\\eps(x)\\to {8\\pi+ \\lambda\\over2} G(x,\\xi) \\ \\hbox{uniformly on compact sets of $\\Omega\\setminus\\{\\xi\\}$}$$ as $\\eps$ goes to zero. Here $G$ denotes Green's function of Dirichlet Laplacian in $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3768","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"}