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We show that for $\\lambda=\\lambda^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\\geq 13$ for $p$ large enough and $1-C_{0}r^{\\frac{4}{p+1}}\\l"},"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":"1009.2546","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-14T00:49:31Z","cross_cats_sorted":[],"title_canon_sha256":"45996384a62f1e17e4f8d37ce43d1b9dc89d3b44efb5f43bf721177b3052b9ad","abstract_canon_sha256":"1547d6e587655819ccc1b60e905b401b2275f6d29e6d513ac3e984c860a94d88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:03.215952Z","signature_b64":"SY52XXXiGAh7jDv7Oqku55gUBJRwHXB8utYCcN3yim1EoKh70VWB08W+T+b0xP9w6DwVfbzx8XclQHgsIZMgAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef93b586bc8850f1e0e1ab3234a4dc981bed467b079916b13360514c8e9f7fe7","last_reissued_at":"2026-05-18T04:17:03.215482Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:03.215482Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Properties of the extremal solution for a fourth-order elliptic problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Baishun Lai, Zhuoran Du","submitted_at":"2010-09-14T00:49:31Z","abstract_excerpt":"Let $\\lambda^{*}>0$ denote the largest possible value of $\\lambda$ such that $$ \\{{array}{lllllll} \\Delta^{2}u=\\frac{\\lambda}{(1-u)^{p}} & \\{in}\\ \\ B, 0<u\\leq 1 & \\{in}\\ \\ B, u=\\frac{\\partial u}{\\partial n} =0 & \\{on}\\ \\ \\partial B. {array} . $$ has a solution, where $B$ is the unit ball in $R^{n}$ centered at the origin, $p>1$ and $n$ is the exterior unit normal vector. We show that for $\\lambda=\\lambda^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\\geq 13$ for $p$ large enough and $1-C_{0}r^{\\frac{4}{p+1}}\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2546","kind":"arxiv","version":3},"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":"1009.2546","created_at":"2026-05-18T04:17:03.215544+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.2546v3","created_at":"2026-05-18T04:17:03.215544+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2546","created_at":"2026-05-18T04:17:03.215544+00:00"},{"alias_kind":"pith_short_12","alias_value":"56J3LBV4RBIP","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"56J3LBV4RBIPDYHB","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"56J3LBV4","created_at":"2026-05-18T12:26:04.259169+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/56J3LBV4RBIPDYHBVMZDJJG4TA","json":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA.json","graph_json":"https://pith.science/api/pith-number/56J3LBV4RBIPDYHBVMZDJJG4TA/graph.json","events_json":"https://pith.science/api/pith-number/56J3LBV4RBIPDYHBVMZDJJG4TA/events.json","paper":"https://pith.science/paper/56J3LBV4"},"agent_actions":{"view_html":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA","download_json":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA.json","view_paper":"https://pith.science/paper/56J3LBV4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.2546&json=true","fetch_graph":"https://pith.science/api/pith-number/56J3LBV4RBIPDYHBVMZDJJG4TA/graph.json","fetch_events":"https://pith.science/api/pith-number/56J3LBV4RBIPDYHBVMZDJJG4TA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA/action/storage_attestation","attest_author":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA/action/author_attestation","sign_citation":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA/action/citation_signature","submit_replication":"https://pith.science/pith/56J3LBV4RBIPDYHBVMZDJJG4TA/action/replication_record"}},"created_at":"2026-05-18T04:17:03.215544+00:00","updated_at":"2026-05-18T04:17:03.215544+00:00"}