{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2001:JBVDXI365UUHYNIYCGTBPSB3WQ","short_pith_number":"pith:JBVDXI36","schema_version":"1.0","canonical_sha256":"486a3ba37eed287c351811a617c83bb415915fd8af9894557c204d03e1a81f48","source":{"kind":"arxiv","id":"math/0106231","version":1},"attestation_state":"computed","paper":{"title":"Some Liouville Theorems for the p-Laplacian","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"F. Demengel, I. Birindelli","submitted_at":"2001-06-27T12:40:05Z","abstract_excerpt":"We present several Liouville type results for the $p$-Laplacian in $\\R^N$. Suppose that\n $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\\gamma\\ {\\rm for}\\ |x|\\ {\\rm large},\\ a>0\\ {\\rm and}\\ \\gamma> -p. $$ We obtain the following non -existence result:\n  1) Suppose that $N>p>1$, and $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R^N)$ is a nonnegative weak solution of $ - {\\rm div} (|\\nabla u|^{p-2 }\\nabla u) \\geq h(x) u^q \\;\\;\\mbox{in }\\; \\R^N $ . Suppose that $p-1< q\\leq {(N+\\gamma)(p-1)\\over N-p}$ then $u\\equiv 0$.\n  2) Let $N\\leq p$. If $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R"},"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":"math/0106231","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2001-06-27T12:40:05Z","cross_cats_sorted":[],"title_canon_sha256":"d3aa05f49e6839621e0272e6f8a9cdb899e3c12982445af950304cc5a5c50cca","abstract_canon_sha256":"599daa8b7e9633d2889acb45d688197cbbd7782ceaf0419ddf12aa0fd5ab2cfb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:37.696159Z","signature_b64":"JioTsqamktAUeAyG8EtKL1heKHCGyVN25Galin+2sVaPcz+YWCPx+sogeTzt/4ta842IxjIBK4OdKv4Wxw+ACA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"486a3ba37eed287c351811a617c83bb415915fd8af9894557c204d03e1a81f48","last_reissued_at":"2026-05-18T01:05:37.695560Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:37.695560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Liouville Theorems for the p-Laplacian","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"F. Demengel, I. Birindelli","submitted_at":"2001-06-27T12:40:05Z","abstract_excerpt":"We present several Liouville type results for the $p$-Laplacian in $\\R^N$. Suppose that\n $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\\gamma\\ {\\rm for}\\ |x|\\ {\\rm large},\\ a>0\\ {\\rm and}\\ \\gamma> -p. $$ We obtain the following non -existence result:\n  1) Suppose that $N>p>1$, and $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R^N)$ is a nonnegative weak solution of $ - {\\rm div} (|\\nabla u|^{p-2 }\\nabla u) \\geq h(x) u^q \\;\\;\\mbox{in }\\; \\R^N $ . Suppose that $p-1< q\\leq {(N+\\gamma)(p-1)\\over N-p}$ then $u\\equiv 0$.\n  2) Let $N\\leq p$. If $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0106231","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":"math/0106231","created_at":"2026-05-18T01:05:37.695665+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0106231v1","created_at":"2026-05-18T01:05:37.695665+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0106231","created_at":"2026-05-18T01:05:37.695665+00:00"},{"alias_kind":"pith_short_12","alias_value":"JBVDXI365UUH","created_at":"2026-05-18T12:25:50.254431+00:00"},{"alias_kind":"pith_short_16","alias_value":"JBVDXI365UUHYNIY","created_at":"2026-05-18T12:25:50.254431+00:00"},{"alias_kind":"pith_short_8","alias_value":"JBVDXI36","created_at":"2026-05-18T12:25:50.254431+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/JBVDXI365UUHYNIYCGTBPSB3WQ","json":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ.json","graph_json":"https://pith.science/api/pith-number/JBVDXI365UUHYNIYCGTBPSB3WQ/graph.json","events_json":"https://pith.science/api/pith-number/JBVDXI365UUHYNIYCGTBPSB3WQ/events.json","paper":"https://pith.science/paper/JBVDXI36"},"agent_actions":{"view_html":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ","download_json":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ.json","view_paper":"https://pith.science/paper/JBVDXI36","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0106231&json=true","fetch_graph":"https://pith.science/api/pith-number/JBVDXI365UUHYNIYCGTBPSB3WQ/graph.json","fetch_events":"https://pith.science/api/pith-number/JBVDXI365UUHYNIYCGTBPSB3WQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ/action/storage_attestation","attest_author":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ/action/author_attestation","sign_citation":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ/action/citation_signature","submit_replication":"https://pith.science/pith/JBVDXI365UUHYNIYCGTBPSB3WQ/action/replication_record"}},"created_at":"2026-05-18T01:05:37.695665+00:00","updated_at":"2026-05-18T01:05:37.695665+00:00"}