{"paper":{"title":"A Topological Degree Counting for some Liouville Systems of Mean Field Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Lei Zhang","submitted_at":"2010-09-01T19:50:01Z","abstract_excerpt":"Let $A=(a_{ij})_{n\\times n}$ be an invertible matrix and $A^{-1}=(a^{ij})_{n\\times n}$ be the inverse of $A$. In this paper, we consider the generalized Liouville system: \\label{abeq1} \\Delta_g u_i+\\sum_{j=1}^n a_{ij}\\rho_j(\\frac{h_j e^{u_j}}{\\int h_j e^{u_j}}-1)=0\\quad\\text{in \\,}M,  where $0< h_j\\in C^1(M)$ and $\\rho_j\\in \\mathbb R^+$, and prove that, under the assumptions of $(H_1)$ and $(H_2)$\\,(see Introduction), the Leray-Schauder degree of \\eqref{abeq1} is equal to \\frac{(-\\chi(M)+1)... (-\\chi(M)+N)}{N!} if $\\rho=(\\rho_1,..., \\rho_n)$ satisfies 8\\pi N\\sum_{i=1}^n\\rho_i<\\sum_{1\\leq i,j\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0259","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"}