{"paper":{"title":"On the Topological degree of the Mean field equation with two parameters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aleks Jevnikar, Juncheng Wei, Wen Yang","submitted_at":"2016-02-10T12:45:34Z","abstract_excerpt":"We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$\n  - \\Delta u = \\rho_1 \\left( \\frac{h_1 \\,e^{u}}{\\int_M\n  h_1 \\,e^{u} } - \\frac{1}{|M|} \\right) - \\rho_2 \\left( \\frac{h_2 \\,e^{-u}}{\\int_M\n  h_2 \\,e^{-u} } - \\frac{1}{|M|} \\right), $$ which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h_1, h_2$ are smooth positive functions and $\\rho_1, \\rho_2$ are two positive parameters.\n  We start by proving a concentration phenomena for the above equation, which leads to a-priori bound fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03354","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"}