{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:3VSAHLBSZCPM7JXUHGY63KAEPP","short_pith_number":"pith:3VSAHLBS","schema_version":"1.0","canonical_sha256":"dd6403ac32c89ecfa6f439b1eda8047bcb0c3226edcfca67680c948d167aeeae","source":{"kind":"arxiv","id":"1511.03948","version":1},"attestation_state":"computed","paper":{"title":"Biharmonic equation with singular nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"G. Warnault, J. Giacomoni, S. Prashanth","submitted_at":"2015-11-12T16:26:14Z","abstract_excerpt":"We consider the following problem: \\begin{eqnarray*} ( P)\\qquad \\displaystyle\\left\\{\\begin{array} {ll}\n  & \\Delta^2 u\n  = K(x)u^{-\\alpha}\n  \\quad \\mbox{ in }\\,\\Omega , \\\\ &u> 0\\quad \\mbox{ in }\\,\\Omega, \\;\\;u\\vert_{\\partial\\Omega}=0, \\,\\Delta u\\vert_{\\partial\\Omega} = 0. \\end{array}\\right.\n  \\end{eqnarray*} We prove the main existence result:\n  Assume that $\\alpha+\\beta<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \\begin{eqnarray}\\label{behaviour-bound} c_1 \\rho(x)\\leq u(x)\\leq c_2 \\rho(x) \\end{eqnarray}\n  where $\\rho(x)=d(x,\\partial\\Ome"},"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":"1511.03948","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-12T16:26:14Z","cross_cats_sorted":[],"title_canon_sha256":"41fe648df200982ee54c31b7823f95702d1960d11a73d18454694c0a20cd96b8","abstract_canon_sha256":"86854ac151ed9d83ae2ca032d729ea84d2672cd09ca6fc47491ce599b49c864f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:06.381467Z","signature_b64":"QPgGcZSBifoF6hreWyfZbMENZ0of5ANYqyjYX0RwElAV5GbhCt0BIamcaqdeR4alaklprcod+qoLlxkKQtUGAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd6403ac32c89ecfa6f439b1eda8047bcb0c3226edcfca67680c948d167aeeae","last_reissued_at":"2026-05-18T01:27:06.380741Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:06.380741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Biharmonic equation with singular nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"G. Warnault, J. Giacomoni, S. Prashanth","submitted_at":"2015-11-12T16:26:14Z","abstract_excerpt":"We consider the following problem: \\begin{eqnarray*} ( P)\\qquad \\displaystyle\\left\\{\\begin{array} {ll}\n  & \\Delta^2 u\n  = K(x)u^{-\\alpha}\n  \\quad \\mbox{ in }\\,\\Omega , \\\\ &u> 0\\quad \\mbox{ in }\\,\\Omega, \\;\\;u\\vert_{\\partial\\Omega}=0, \\,\\Delta u\\vert_{\\partial\\Omega} = 0. \\end{array}\\right.\n  \\end{eqnarray*} We prove the main existence result:\n  Assume that $\\alpha+\\beta<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \\begin{eqnarray}\\label{behaviour-bound} c_1 \\rho(x)\\leq u(x)\\leq c_2 \\rho(x) \\end{eqnarray}\n  where $\\rho(x)=d(x,\\partial\\Ome"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03948","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":"1511.03948","created_at":"2026-05-18T01:27:06.380859+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.03948v1","created_at":"2026-05-18T01:27:06.380859+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.03948","created_at":"2026-05-18T01:27:06.380859+00:00"},{"alias_kind":"pith_short_12","alias_value":"3VSAHLBSZCPM","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"3VSAHLBSZCPM7JXU","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"3VSAHLBS","created_at":"2026-05-18T12:29:02.477457+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/3VSAHLBSZCPM7JXUHGY63KAEPP","json":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP.json","graph_json":"https://pith.science/api/pith-number/3VSAHLBSZCPM7JXUHGY63KAEPP/graph.json","events_json":"https://pith.science/api/pith-number/3VSAHLBSZCPM7JXUHGY63KAEPP/events.json","paper":"https://pith.science/paper/3VSAHLBS"},"agent_actions":{"view_html":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP","download_json":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP.json","view_paper":"https://pith.science/paper/3VSAHLBS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.03948&json=true","fetch_graph":"https://pith.science/api/pith-number/3VSAHLBSZCPM7JXUHGY63KAEPP/graph.json","fetch_events":"https://pith.science/api/pith-number/3VSAHLBSZCPM7JXUHGY63KAEPP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP/action/storage_attestation","attest_author":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP/action/author_attestation","sign_citation":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP/action/citation_signature","submit_replication":"https://pith.science/pith/3VSAHLBSZCPM7JXUHGY63KAEPP/action/replication_record"}},"created_at":"2026-05-18T01:27:06.380859+00:00","updated_at":"2026-05-18T01:27:06.380859+00:00"}