{"paper":{"title":"Classification of Radial Solutions to Liouville Systems with Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Lei Zhang","submitted_at":"2013-02-15T20:25:07Z","abstract_excerpt":"Let $A=(a_{ij})_{n\\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: $$\\{{array}{ll} \\Delta u_i+\\sum_{j=1}^n a_{ij}|x|^{\\beta_j}e^{u_j(x)}=0,\\quad \\mathbb R^2, \\quad i=1,...,n \\int_{\\mathbb R^2}|x|^{\\beta_i}e^{u_i(x)}dx<\\infty, \\quad i=1,...,n {array}. $$ where $\\beta_1,...,\\beta_n$ are constants greater than -2. If all $\\beta_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.3866","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"}