{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:GDDVQP77JWFNUSXQRI2JDR2VFK","short_pith_number":"pith:GDDVQP77","schema_version":"1.0","canonical_sha256":"30c7583fff4d8ada4af08a3491c7552ab563f9a0c73d605b5687fadd943b4ef0","source":{"kind":"arxiv","id":"2012.11023","version":1},"attestation_state":"computed","paper":{"title":"Liouville type theorems for stable solutions of elliptic system involving the Grushin operator","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-12-20T21:05:31Z","abstract_excerpt":"We examine the degenerate elliptic system\n  $$-\\Delta_{s} u = v^p, \\quad -\\Delta_{s} v= u^\\theta, \\quad u,v>0 \\quad\\mbox{in }\\; \\mathbb{R}^N=\\mathbb{R}^{N_1}\\times \\mathbb{R}^{N_2}, \\quad\\mbox{where }\\;\\;\\;\\; s \\geq 0\\;\\; \\mbox{and} \\;\\;p,\\theta >0.$$ We prove that the system has no smooth stable solution provided $p,\\theta >0$ and $N_s< 2 + \\alpha + \\beta,$ where\n  $$\\alpha = \\frac{2(p+1)}{p\\theta - 1} \\quad\\mbox{and} \\quad \\beta = \\frac{2(\\theta +1)}{p\\theta - 1}.$$\n  This result is an extension of some result in \\cite{ MY}. In particular, we establish a new the integral estimate for $u$ and"},"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":"2012.11023","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2020-12-20T21:05:31Z","cross_cats_sorted":[],"title_canon_sha256":"8eacbf7d8f398cf2653a0dacd1886b12408736ca168b54936d777683abd1aaa3","abstract_canon_sha256":"6dee98b93e00fb88080f7bd0b83d37cc7f40774716c672546e5b7bee93f94287"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:00:57.811585Z","signature_b64":"/vnvBn8tfSCl5evnmfDUMlZZfZo0oh4XcX/icjffOASRxkSZ0zb0y4hhsCUMqgDjnyQEix2+W4TjyTDiMQBSDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30c7583fff4d8ada4af08a3491c7552ab563f9a0c73d605b5687fadd943b4ef0","last_reissued_at":"2026-07-05T02:00:57.811161Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:00:57.811161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Liouville type theorems for stable solutions of elliptic system involving the Grushin operator","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-12-20T21:05:31Z","abstract_excerpt":"We examine the degenerate elliptic system\n  $$-\\Delta_{s} u = v^p, \\quad -\\Delta_{s} v= u^\\theta, \\quad u,v>0 \\quad\\mbox{in }\\; \\mathbb{R}^N=\\mathbb{R}^{N_1}\\times \\mathbb{R}^{N_2}, \\quad\\mbox{where }\\;\\;\\;\\; s \\geq 0\\;\\; \\mbox{and} \\;\\;p,\\theta >0.$$ We prove that the system has no smooth stable solution provided $p,\\theta >0$ and $N_s< 2 + \\alpha + \\beta,$ where\n  $$\\alpha = \\frac{2(p+1)}{p\\theta - 1} \\quad\\mbox{and} \\quad \\beta = \\frac{2(\\theta +1)}{p\\theta - 1}.$$\n  This result is an extension of some result in \\cite{ MY}. In particular, we establish a new the integral estimate for $u$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.11023","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2012.11023/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":"2012.11023","created_at":"2026-07-05T02:00:57.811214+00:00"},{"alias_kind":"arxiv_version","alias_value":"2012.11023v1","created_at":"2026-07-05T02:00:57.811214+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.11023","created_at":"2026-07-05T02:00:57.811214+00:00"},{"alias_kind":"pith_short_12","alias_value":"GDDVQP77JWFN","created_at":"2026-07-05T02:00:57.811214+00:00"},{"alias_kind":"pith_short_16","alias_value":"GDDVQP77JWFNUSXQ","created_at":"2026-07-05T02:00:57.811214+00:00"},{"alias_kind":"pith_short_8","alias_value":"GDDVQP77","created_at":"2026-07-05T02:00:57.811214+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/GDDVQP77JWFNUSXQRI2JDR2VFK","json":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK.json","graph_json":"https://pith.science/api/pith-number/GDDVQP77JWFNUSXQRI2JDR2VFK/graph.json","events_json":"https://pith.science/api/pith-number/GDDVQP77JWFNUSXQRI2JDR2VFK/events.json","paper":"https://pith.science/paper/GDDVQP77"},"agent_actions":{"view_html":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK","download_json":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK.json","view_paper":"https://pith.science/paper/GDDVQP77","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2012.11023&json=true","fetch_graph":"https://pith.science/api/pith-number/GDDVQP77JWFNUSXQRI2JDR2VFK/graph.json","fetch_events":"https://pith.science/api/pith-number/GDDVQP77JWFNUSXQRI2JDR2VFK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK/action/storage_attestation","attest_author":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK/action/author_attestation","sign_citation":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK/action/citation_signature","submit_replication":"https://pith.science/pith/GDDVQP77JWFNUSXQRI2JDR2VFK/action/replication_record"}},"created_at":"2026-07-05T02:00:57.811214+00:00","updated_at":"2026-07-05T02:00:57.811214+00:00"}