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We first prove a Liouville theorem (Theorem \\ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \\ref{Thm1} and \\ref{Thm2}"},"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":"1808.06609","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-08-20T12:48:53Z","cross_cats_sorted":[],"title_canon_sha256":"7fa13a44d141556ace7604a75fd66bd3e46df332ff8b0312b81de8150c79925f","abstract_canon_sha256":"d3245e1833b6683ebdd2c372bd8792fa550c117123cde7e33e0a0a8aaa42f527"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:19.488483Z","signature_b64":"LHOcW4ia1bMVODnGl9yLNrjG7UHH7IHuwtJLHzuOjRdcaF14rCmAczy2RqjB114nhlo3vTH0eQgP0YoKmlb5Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c70e9a178255511f4159bc2efa638a6dec8399ea31a8772f0863af82b3dba09","last_reissued_at":"2026-05-17T23:46:19.487965Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:19.487965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-H\\'{e}non 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":"2018-08-20T12:48:53Z","abstract_excerpt":"In this paper, we consider the critical order Hardy-H\\'{e}non equations \\begin{equation*}\n  (-\\Delta)^{\\frac{n}{2}}u(x)=\\frac{u^{p}(x)}{|x|^{a}}, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, x \\, \\in \\,\\, \\mathbb{R}^{n}, \\end{equation*} where $n\\geq4$ is even, $-\\infty<a<n$, and $1<p<+\\infty$. We first prove a Liouville theorem (Theorem \\ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \\ref{Thm1} and \\ref{Thm2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06609","kind":"arxiv","version":4},"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":"1808.06609","created_at":"2026-05-17T23:46:19.488049+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.06609v4","created_at":"2026-05-17T23:46:19.488049+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06609","created_at":"2026-05-17T23:46:19.488049+00:00"},{"alias_kind":"pith_short_12","alias_value":"HRYOTILYEVKR","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"HRYOTILYEVKRD5AV","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"HRYOTILY","created_at":"2026-05-18T12:32:28.185984+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/HRYOTILYEVKRD5AVTPBO7JRYU3","json":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3.json","graph_json":"https://pith.science/api/pith-number/HRYOTILYEVKRD5AVTPBO7JRYU3/graph.json","events_json":"https://pith.science/api/pith-number/HRYOTILYEVKRD5AVTPBO7JRYU3/events.json","paper":"https://pith.science/paper/HRYOTILY"},"agent_actions":{"view_html":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3","download_json":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3.json","view_paper":"https://pith.science/paper/HRYOTILY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.06609&json=true","fetch_graph":"https://pith.science/api/pith-number/HRYOTILYEVKRD5AVTPBO7JRYU3/graph.json","fetch_events":"https://pith.science/api/pith-number/HRYOTILYEVKRD5AVTPBO7JRYU3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3/action/storage_attestation","attest_author":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3/action/author_attestation","sign_citation":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3/action/citation_signature","submit_replication":"https://pith.science/pith/HRYOTILYEVKRD5AVTPBO7JRYU3/action/replication_record"}},"created_at":"2026-05-17T23:46:19.488049+00:00","updated_at":"2026-05-17T23:46:19.488049+00:00"}