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We prove the existence of $\\lambda^{*}>0$ such that for $\\lambda\\in (0,\\lambda^{*})$ there exists a minimal (classical) solution $\\underline{u}_{\\lambda}$, which satisfies $0<\\underline{u}_{\\lambda}<1$. In the extremal case $\\lambda=\\lambda^{*}$, we prove th"},"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":"1101.3904","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-01-20T13:52:52Z","cross_cats_sorted":[],"title_canon_sha256":"23fc90b61ed966edd7b721dfc26d23d5117720c929bde9da47ace6804375e38a","abstract_canon_sha256":"49ab76728ada75c2883afdec1a1b34bdfa813f95fe067171a818e2c734af9b4e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:20.850431Z","signature_b64":"KRlr7WGACI3gSM0RC/NabGceDcVAId0ugYbCxwNYO2PatAVKVIaycYGzD+P8pC3BtFKxIEiTfNUAvrfmvmgNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3e72a2995455f3900966e5d6e45efaeb15b386221fcf922318fc4487fe59a50","last_reissued_at":"2026-05-18T04:31:20.849908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:20.849908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some remarks on biharmonic elliptic problems with a singular nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Baishun Lai","submitted_at":"2011-01-20T13:52:52Z","abstract_excerpt":"We study the following semilinear biharmonic equation $$ \\left\\{\\begin{array}{lllllll} \\Delta^{2}u=\\frac{\\lambda}{1-u}, &\\quad \\mbox{in}\\quad \\B, u=\\frac{\\partial u}{\\partial n}=0, &\\quad \\mbox{on}\\quad \\partial\\B, \\end{array} \\right. %\\eqno(M_{\\lambda}) $$ where $\\B$ is the unit ball in $\\R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $\\lambda^{*}>0$ such that for $\\lambda\\in (0,\\lambda^{*})$ there exists a minimal (classical) solution $\\underline{u}_{\\lambda}$, which satisfies $0<\\underline{u}_{\\lambda}<1$. In the extremal case $\\lambda=\\lambda^{*}$, we prove th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3904","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":"1101.3904","created_at":"2026-05-18T04:31:20.849976+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.3904v1","created_at":"2026-05-18T04:31:20.849976+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3904","created_at":"2026-05-18T04:31:20.849976+00:00"},{"alias_kind":"pith_short_12","alias_value":"6PTSUKMVIVPT","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"6PTSUKMVIVPTSAEW","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"6PTSUKMV","created_at":"2026-05-18T12:26:22.705136+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/6PTSUKMVIVPTSAEWNZOW4RPPV2","json":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2.json","graph_json":"https://pith.science/api/pith-number/6PTSUKMVIVPTSAEWNZOW4RPPV2/graph.json","events_json":"https://pith.science/api/pith-number/6PTSUKMVIVPTSAEWNZOW4RPPV2/events.json","paper":"https://pith.science/paper/6PTSUKMV"},"agent_actions":{"view_html":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2","download_json":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2.json","view_paper":"https://pith.science/paper/6PTSUKMV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.3904&json=true","fetch_graph":"https://pith.science/api/pith-number/6PTSUKMVIVPTSAEWNZOW4RPPV2/graph.json","fetch_events":"https://pith.science/api/pith-number/6PTSUKMVIVPTSAEWNZOW4RPPV2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2/action/storage_attestation","attest_author":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2/action/author_attestation","sign_citation":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2/action/citation_signature","submit_replication":"https://pith.science/pith/6PTSUKMVIVPTSAEWNZOW4RPPV2/action/replication_record"}},"created_at":"2026-05-18T04:31:20.849976+00:00","updated_at":"2026-05-18T04:31:20.849976+00:00"}