{"paper":{"title":"The non-local mean-field equation on an interval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder, Azahara DelaTorre, Luca Martinazzi, Yannick Sire","submitted_at":"2018-12-05T18:57:22Z","abstract_excerpt":"We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-\\Delta)^\\frac{1}{2} u=\\rho\\frac{e^{u}}{\\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $\\rho <2\\pi$. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.02165","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"}