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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, in which case $u^{*}(x)\\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":"1101.3903","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-01-20T13:47:59Z","cross_cats_sorted":[],"title_canon_sha256":"1df991400b5d012da771f66e31c7cce9692150ddb2136e739175758167ae07ce","abstract_canon_sha256":"3dddb2ab4e4558818d6fbf10ee7ee8071bc3b1e7b6227dd618c264bace376861"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:13.513084Z","signature_b64":"AcFGS3ADwwOtSeRQqGLVvNW9uFd7K0hYQthf9xqO45cJmaJxDIcd6nsXcX+4SbenHx/8FHLAGWiZ8NQ9A4XsCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f76ddf0846ee5aeeea22e6c3ab7a056d040c73a04932cfea415d5bbe20b87d9","last_reissued_at":"2026-05-18T04:17:13.512434Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:13.512434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Baishun Lai, Yinghui Zhang, Zhengxiang Yan","submitted_at":"2011-01-20T13:47:59Z","abstract_excerpt":"Let $\\lambda^{*}>0$ denote the largest possible value of $\\lambda$ such that $$ \\{{array}{lllllll} \\Delta^{2}u=\\lambda(1+u)^{p} & {in}\\ \\ \\B, %0<u\\leq 1 & {in}\\ \\ \\B, u=\\frac{\\partial u}{\\partial n} =0 & {on}\\ \\ \\partial \\B {array}. $$ has a solution, where $\\B$ is the unit ball in $R^{n}$ centered at the origin, $p>\\frac{n+4}{n-4}$ 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, in which case $u^{*}(x)\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3903","kind":"arxiv","version":2},"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":"1101.3903","created_at":"2026-05-18T04:17:13.512534+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.3903v2","created_at":"2026-05-18T04:17:13.512534+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3903","created_at":"2026-05-18T04:17:13.512534+00:00"},{"alias_kind":"pith_short_12","alias_value":"H53N34EEN3S2","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"H53N34EEN3S253VC","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"H53N34EE","created_at":"2026-05-18T12:26:30.835961+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/H53N34EEN3S253VCFZWDVN5AK3","json":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3.json","graph_json":"https://pith.science/api/pith-number/H53N34EEN3S253VCFZWDVN5AK3/graph.json","events_json":"https://pith.science/api/pith-number/H53N34EEN3S253VCFZWDVN5AK3/events.json","paper":"https://pith.science/paper/H53N34EE"},"agent_actions":{"view_html":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3","download_json":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3.json","view_paper":"https://pith.science/paper/H53N34EE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.3903&json=true","fetch_graph":"https://pith.science/api/pith-number/H53N34EEN3S253VCFZWDVN5AK3/graph.json","fetch_events":"https://pith.science/api/pith-number/H53N34EEN3S253VCFZWDVN5AK3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3/action/storage_attestation","attest_author":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3/action/author_attestation","sign_citation":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3/action/citation_signature","submit_replication":"https://pith.science/pith/H53N34EEN3S253VCFZWDVN5AK3/action/replication_record"}},"created_at":"2026-05-18T04:17:13.512534+00:00","updated_at":"2026-05-18T04:17:13.512534+00:00"}