{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:E4MJRGLF5DGMMTG4WW6TUOW3FT","short_pith_number":"pith:E4MJRGLF","schema_version":"1.0","canonical_sha256":"2718989965e8ccc64cdcb5bd3a3adb2cc069ee7823eb624f6f3ac87c8da6cf10","source":{"kind":"arxiv","id":"1811.00881","version":4},"attestation_state":"computed","paper":{"title":"Liouville type theorem for critical order H\\'{e}non-Lane-Emden type equations on a half space and its applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai","submitted_at":"2018-11-01T02:31:28Z","abstract_excerpt":"In this paper, we are concerned with the critical order H\\'{e}non-Lane-Emden type equations with Navier boundary condition on a half space $\\mathbb{R}^n_+$: \\begin{equation}\\label{NPDE0}\\\\\\begin{cases} (-\\Delta)^{\\frac{n}{2}} u(x)=f(x,u(x)),\\  u(x)\\geq0,\\ x\\in\\mathbb{R}^{n}_+, \\\\ u=(-\\Delta)u = \\cdots = (-\\Delta)^{\\frac{n}{2}-1}u = 0,\\ x\\in\\partial\\mathbb{R}^{n}_+, \\end{cases}\\end{equation} where $u\\in C^{n}(\\mathbb{R}^{n}_+)\\cap C^{n-2}(\\overline{\\mathbb{R}^{n}_+})$ and $n\\geq2$ is even. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $0\\leq a<\\infty$ and $1<p<\\infty$. We prove "},"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":"1811.00881","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-01T02:31:28Z","cross_cats_sorted":[],"title_canon_sha256":"d639e2aad7c534a3e9cf6b50c70db86b4aa17fc157c2392a4e73a5ff69282a6b","abstract_canon_sha256":"266a8f66ec672190d4f6cf8e0fabf10e9e422770222294805f253855ec5a9b74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:11:08.571498Z","signature_b64":"HDzRJSsRUU7l/GPyH4/BWi7W6GN95bPnJ6Rt7uFCVMh9jaW5PEsk33wfkJstR3YYxCWw/5iRIZQCY40Base/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2718989965e8ccc64cdcb5bd3a3adb2cc069ee7823eb624f6f3ac87c8da6cf10","last_reissued_at":"2026-07-05T03:11:08.571089Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:11:08.571089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Liouville type theorem for critical order H\\'{e}non-Lane-Emden type equations on a half space and its applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guolin Qin, Wei Dai","submitted_at":"2018-11-01T02:31:28Z","abstract_excerpt":"In this paper, we are concerned with the critical order H\\'{e}non-Lane-Emden type equations with Navier boundary condition on a half space $\\mathbb{R}^n_+$: \\begin{equation}\\label{NPDE0}\\\\\\begin{cases} (-\\Delta)^{\\frac{n}{2}} u(x)=f(x,u(x)),\\  u(x)\\geq0,\\ x\\in\\mathbb{R}^{n}_+, \\\\ u=(-\\Delta)u = \\cdots = (-\\Delta)^{\\frac{n}{2}-1}u = 0,\\ x\\in\\partial\\mathbb{R}^{n}_+, \\end{cases}\\end{equation} where $u\\in C^{n}(\\mathbb{R}^{n}_+)\\cap C^{n-2}(\\overline{\\mathbb{R}^{n}_+})$ and $n\\geq2$ is even. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $0\\leq a<\\infty$ and $1<p<\\infty$. We prove "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00881","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1811.00881/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1811.00881","created_at":"2026-07-05T03:11:08.571147+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.00881v4","created_at":"2026-07-05T03:11:08.571147+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.00881","created_at":"2026-07-05T03:11:08.571147+00:00"},{"alias_kind":"pith_short_12","alias_value":"E4MJRGLF5DGM","created_at":"2026-07-05T03:11:08.571147+00:00"},{"alias_kind":"pith_short_16","alias_value":"E4MJRGLF5DGMMTG4","created_at":"2026-07-05T03:11:08.571147+00:00"},{"alias_kind":"pith_short_8","alias_value":"E4MJRGLF","created_at":"2026-07-05T03:11:08.571147+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1909.00492","citing_title":"Classification of nonnegative solutions to static Schr\\\"{o}dinger-Hartree-Maxwell type equations","ref_index":31,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT","json":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT.json","graph_json":"https://pith.science/api/pith-number/E4MJRGLF5DGMMTG4WW6TUOW3FT/graph.json","events_json":"https://pith.science/api/pith-number/E4MJRGLF5DGMMTG4WW6TUOW3FT/events.json","paper":"https://pith.science/paper/E4MJRGLF"},"agent_actions":{"view_html":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT","download_json":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT.json","view_paper":"https://pith.science/paper/E4MJRGLF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.00881&json=true","fetch_graph":"https://pith.science/api/pith-number/E4MJRGLF5DGMMTG4WW6TUOW3FT/graph.json","fetch_events":"https://pith.science/api/pith-number/E4MJRGLF5DGMMTG4WW6TUOW3FT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT/action/storage_attestation","attest_author":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT/action/author_attestation","sign_citation":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT/action/citation_signature","submit_replication":"https://pith.science/pith/E4MJRGLF5DGMMTG4WW6TUOW3FT/action/replication_record"}},"created_at":"2026-07-05T03:11:08.571147+00:00","updated_at":"2026-07-05T03:11:08.571147+00:00"}