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We prove Liouville theorem for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \\ref{Thm0}), that is, the unique nonnegative solution is $u\\equiv0$. Our result seems to be the first Liouville theorem on the critical order equations in higher dimensions ($n\\geq3$)."},"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":"1808.01581","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-08-05T08:38:42Z","cross_cats_sorted":[],"title_canon_sha256":"5f6a968661ea4d3547c00b8672d797f8e0b7b7c337c0c0c965ed2da98de7fffa","abstract_canon_sha256":"853054ad3c5e7b1c2d0068e458f5ec0abec57f07822570381c789472f62a7933"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:52.537301Z","signature_b64":"WyUHU3Bwd1Qwof7u+KEHBQkifh0Mp2BmwIjgVCkfu/21a9loRjBG0gY+aRlvq/EE4JFR6mNiRS6GfKbu4tjCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81df77aaed07994385255296a21455b87396483b8ffe77463bbe0892fa09a256","last_reissued_at":"2026-05-18T00:08:52.536576Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:52.536576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Liouville type theorem for critical order Lane-Emden-Hardy equations in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai, Wenxiong Chen","submitted_at":"2018-08-05T08:38:42Z","abstract_excerpt":"In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \\begin{equation*}\n  (-\\Delta)^{\\frac{n}{2}}u(x)=\\frac{u^{p}(x)}{|x|^{a}} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{in} \\,\\,\\, \\mathbb{R}^{n} \\end{equation*} with $n\\geq4$ is even, $0\\leq a<n$ and $1<p<+\\infty$. We prove Liouville theorem for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \\ref{Thm0}), that is, the unique nonnegative solution is $u\\equiv0$. Our result seems to be the first Liouville theorem on the critical order equations in higher dimensions ($n\\geq3$)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01581","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":"1808.01581","created_at":"2026-05-18T00:08:52.536695+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.01581v1","created_at":"2026-05-18T00:08:52.536695+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.01581","created_at":"2026-05-18T00:08:52.536695+00:00"},{"alias_kind":"pith_short_12","alias_value":"QHPXPKXNA6MU","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"QHPXPKXNA6MUHBJF","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"QHPXPKXN","created_at":"2026-05-18T12:32:46.962924+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/QHPXPKXNA6MUHBJFKKLKEFCVXB","json":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB.json","graph_json":"https://pith.science/api/pith-number/QHPXPKXNA6MUHBJFKKLKEFCVXB/graph.json","events_json":"https://pith.science/api/pith-number/QHPXPKXNA6MUHBJFKKLKEFCVXB/events.json","paper":"https://pith.science/paper/QHPXPKXN"},"agent_actions":{"view_html":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB","download_json":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB.json","view_paper":"https://pith.science/paper/QHPXPKXN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.01581&json=true","fetch_graph":"https://pith.science/api/pith-number/QHPXPKXNA6MUHBJFKKLKEFCVXB/graph.json","fetch_events":"https://pith.science/api/pith-number/QHPXPKXNA6MUHBJFKKLKEFCVXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB/action/storage_attestation","attest_author":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB/action/author_attestation","sign_citation":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB/action/citation_signature","submit_replication":"https://pith.science/pith/QHPXPKXNA6MUHBJFKKLKEFCVXB/action/replication_record"}},"created_at":"2026-05-18T00:08:52.536695+00:00","updated_at":"2026-05-18T00:08:52.536695+00:00"}