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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.3866","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-15T20:25:07Z","cross_cats_sorted":[],"title_canon_sha256":"d1d93db3b92f649c35df880325060125be730d344cefb8207459eea3eb3c9b34","abstract_canon_sha256":"4c692f22924ea02a7c6dfae53b8c09bd16e3398b26a7bb41d964198a68f38f8f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:27.188965Z","signature_b64":"pj7krqZ+6E+sB3E9IUC7IuTNZ9iowe9ZChaCiB0Ji+N479d0o6cCnOxmCR7sAr8c0zZs4BfJQCXrGcuE9PsUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1f5ae8cc5d1a9c994a872d1afcd1591414d421b25fce12a2697e24cf727194a","last_reissued_at":"2026-05-18T03:33:27.188208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:27.188208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1302.3866","created_at":"2026-05-18T03:33:27.188342+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.3866v1","created_at":"2026-05-18T03:33:27.188342+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.3866","created_at":"2026-05-18T03:33:27.188342+00:00"},{"alias_kind":"pith_short_12","alias_value":"UH225DGF2GU4","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"UH225DGF2GU4TFFI","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"UH225DGF","created_at":"2026-05-18T12:28:02.375192+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF","json":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF.json","graph_json":"https://pith.science/api/pith-number/UH225DGF2GU4TFFIOLI27TIVSF/graph.json","events_json":"https://pith.science/api/pith-number/UH225DGF2GU4TFFIOLI27TIVSF/events.json","paper":"https://pith.science/paper/UH225DGF"},"agent_actions":{"view_html":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF","download_json":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF.json","view_paper":"https://pith.science/paper/UH225DGF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.3866&json=true","fetch_graph":"https://pith.science/api/pith-number/UH225DGF2GU4TFFIOLI27TIVSF/graph.json","fetch_events":"https://pith.science/api/pith-number/UH225DGF2GU4TFFIOLI27TIVSF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF/action/storage_attestation","attest_author":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF/action/author_attestation","sign_citation":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF/action/citation_signature","submit_replication":"https://pith.science/pith/UH225DGF2GU4TFFIOLI27TIVSF/action/replication_record"}},"created_at":"2026-05-18T03:33:27.188342+00:00","updated_at":"2026-05-18T03:33:27.188342+00:00"}