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We show that in dimension N=5 there are no positive bounded classical solutions of (\\ref{pipe}) provided $ 1 < p < p_4(\\alpha)$."},"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":"1110.2246","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-10-11T01:32:25Z","cross_cats_sorted":[],"title_canon_sha256":"443e72ee2bcf430d4301cfec445b7b2be66c70cd8f744c98c7ceb69b8be744eb","abstract_canon_sha256":"0a83e4bf102082e730e73141f0b9951aa6c5543e27a9723ecaab1b0680a26ed9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:17.242781Z","signature_b64":"KDvvNTJ/YRJas3N2urlmQJw/22mtDhG7JO0iYdR4PElN2MuYfsRnS32GxOLHYePXsjjHV8rlnTEMm4Aeo3sgCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e7567d7bab554574422328577ce7e23923b3198be53bdab75476df473148374","last_reissued_at":"2026-05-18T04:11:17.242214Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:17.242214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Liouville theorem for a fourth order H\\'enon equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Craig Cowan","submitted_at":"2011-10-11T01:32:25Z","abstract_excerpt":"We examine the following fourth order H\\'enon equation\n\\label{pipe} \\Delta^2 u = |x|^\\alpha u^p \\qquad \\text{in}\\ \\IR^N,\nwhere $ 0 < \\alpha$. Define the Hardy-Sobolev exponent $ p_4(\\alpha):= \\frac{N+4 + 2 \\alpha}{N-4}$. We show that in dimension N=5 there are no positive bounded classical solutions of (\\ref{pipe}) provided $ 1 < p < p_4(\\alpha)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2246","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":"1110.2246","created_at":"2026-05-18T04:11:17.242297+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.2246v1","created_at":"2026-05-18T04:11:17.242297+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2246","created_at":"2026-05-18T04:11:17.242297+00:00"},{"alias_kind":"pith_short_12","alias_value":"FZ2WPV52WVKF","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FZ2WPV52WVKFORBC","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FZ2WPV52","created_at":"2026-05-18T12:26:28.662955+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/FZ2WPV52WVKFORBCGKCXPTT6EO","json":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO.json","graph_json":"https://pith.science/api/pith-number/FZ2WPV52WVKFORBCGKCXPTT6EO/graph.json","events_json":"https://pith.science/api/pith-number/FZ2WPV52WVKFORBCGKCXPTT6EO/events.json","paper":"https://pith.science/paper/FZ2WPV52"},"agent_actions":{"view_html":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO","download_json":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO.json","view_paper":"https://pith.science/paper/FZ2WPV52","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.2246&json=true","fetch_graph":"https://pith.science/api/pith-number/FZ2WPV52WVKFORBCGKCXPTT6EO/graph.json","fetch_events":"https://pith.science/api/pith-number/FZ2WPV52WVKFORBCGKCXPTT6EO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO/action/storage_attestation","attest_author":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO/action/author_attestation","sign_citation":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO/action/citation_signature","submit_replication":"https://pith.science/pith/FZ2WPV52WVKFORBCGKCXPTT6EO/action/replication_record"}},"created_at":"2026-05-18T04:11:17.242297+00:00","updated_at":"2026-05-18T04:11:17.242297+00:00"}