{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:6WRHFTRYUGEY6ZBUQQKNOWLQTS","short_pith_number":"pith:6WRHFTRY","schema_version":"1.0","canonical_sha256":"f5a272ce38a1898f64348414d759709c9707d2e914364a8fb58d3efa22f7f761","source":{"kind":"arxiv","id":"1311.6952","version":1},"attestation_state":"computed","paper":{"title":"Radial symmetry of positive solutions involving the fractional Laplacian","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Patricio Felmer, Ying Wang","submitted_at":"2013-11-27T12:33:30Z","abstract_excerpt":"The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\\Delta)^{\\alpha} u=f(u)+g,\\ \\ {\\rm{in}}\\ \\ B_1, \\quad u=0\\ \\ {\\rm in}\\ \\ B_1^c, where $(-\\Delta)^\\alpha$ denotes the fractional Laplacian, $\\alpha\\in(0,1)$, and $B_1$ denotes the open unit ball centered at the origin in $\\R^N$ with $N\\ge2$. The function $f:[0,\\infty)\\to\\R$ is assumed to be locally Lipschitz continuous and $g: B_1\\to\\R$ is radially symmetric and decreasing in $|x|$.\n  I"},"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":"1311.6952","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2013-11-27T12:33:30Z","cross_cats_sorted":[],"title_canon_sha256":"5d9bdbe7048b0448a987f7be986c45486a6b186e1c608176ae8a2dd92c7ab5ab","abstract_canon_sha256":"ff8bb08f7faf2cc4324ee178f45dd40c6ea13e19959a4bffd02efcb8852ba7ce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:04.444543Z","signature_b64":"zZ1hNq0xBXfPJEHOwmShGGWNT4r7FT40Wu3ZS7tTpz605w2WOO9wRhkOWfvLRJoLYNSAGcX9R+8sSksOww90Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5a272ce38a1898f64348414d759709c9707d2e914364a8fb58d3efa22f7f761","last_reissued_at":"2026-05-18T03:06:04.443783Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:04.443783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Radial symmetry of positive solutions involving the fractional Laplacian","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Patricio Felmer, Ying Wang","submitted_at":"2013-11-27T12:33:30Z","abstract_excerpt":"The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\\Delta)^{\\alpha} u=f(u)+g,\\ \\ {\\rm{in}}\\ \\ B_1, \\quad u=0\\ \\ {\\rm in}\\ \\ B_1^c, where $(-\\Delta)^\\alpha$ denotes the fractional Laplacian, $\\alpha\\in(0,1)$, and $B_1$ denotes the open unit ball centered at the origin in $\\R^N$ with $N\\ge2$. The function $f:[0,\\infty)\\to\\R$ is assumed to be locally Lipschitz continuous and $g: B_1\\to\\R$ is radially symmetric and decreasing in $|x|$.\n  I"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6952","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":"1311.6952","created_at":"2026-05-18T03:06:04.443893+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6952v1","created_at":"2026-05-18T03:06:04.443893+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6952","created_at":"2026-05-18T03:06:04.443893+00:00"},{"alias_kind":"pith_short_12","alias_value":"6WRHFTRYUGEY","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6WRHFTRYUGEY6ZBU","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6WRHFTRY","created_at":"2026-05-18T12:27:36.564083+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/6WRHFTRYUGEY6ZBUQQKNOWLQTS","json":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS.json","graph_json":"https://pith.science/api/pith-number/6WRHFTRYUGEY6ZBUQQKNOWLQTS/graph.json","events_json":"https://pith.science/api/pith-number/6WRHFTRYUGEY6ZBUQQKNOWLQTS/events.json","paper":"https://pith.science/paper/6WRHFTRY"},"agent_actions":{"view_html":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS","download_json":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS.json","view_paper":"https://pith.science/paper/6WRHFTRY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6952&json=true","fetch_graph":"https://pith.science/api/pith-number/6WRHFTRYUGEY6ZBUQQKNOWLQTS/graph.json","fetch_events":"https://pith.science/api/pith-number/6WRHFTRYUGEY6ZBUQQKNOWLQTS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS/action/storage_attestation","attest_author":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS/action/author_attestation","sign_citation":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS/action/citation_signature","submit_replication":"https://pith.science/pith/6WRHFTRYUGEY6ZBUQQKNOWLQTS/action/replication_record"}},"created_at":"2026-05-18T03:06:04.443893+00:00","updated_at":"2026-05-18T03:06:04.443893+00:00"}