{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:MANEGKBNNEPLRVAAK7W663DVWJ","short_pith_number":"pith:MANEGKBN","schema_version":"1.0","canonical_sha256":"601a43282d691eb8d40057edef6c75b260452428237f0822a14e24f001c693d0","source":{"kind":"arxiv","id":"2007.03009","version":1},"attestation_state":"computed","paper":{"title":"On the classification of solutions to a weighted elliptic system involving the Grushin operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-07-06T18:53:04Z","abstract_excerpt":"We investigate here the following weighted degenerate elliptic system \\begin{align*} -\\Delta_{s} u = \\Big(1+\\|\\mathbf{x}\\|^{2(s+1)}\\Big)^{\\frac{\\alpha}{2(s+1)}} v^p, \\quad -\\Delta_{s} v = \\Big(1+\\|\\mathbf{x}\\|^{2(s+1)}\\Big)^{\\frac{\\alpha}{2(s+1)}}u^\\theta, \\quad u,v>0\\quad\\mbox{in }\\; \\mathbb{R}^N:=\\mathbb{R}^{N_1}\\times \\mathbb{R}^{N_2}. \\end{align*} where $\\Delta_{s}=\\Delta_{x}+|x|^{2s}\\Delta_{y},$ is the Grushin operator, $s \\geq 0,$ $\\alpha \\geq 0$ and $1<p\\leq\\theta.$ Here $$\\|\\mathbf{x}\\|=\\Big(|x|^{2(s+1)}+|y|^2\\Big)^{\\frac{1}{2(s+1)}}, \\;\\mbox{and}\\;\\; \\mathbf{x}:=(x, y)\\in \\mathbb{R}^N"},"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":"2007.03009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-07-06T18:53:04Z","cross_cats_sorted":[],"title_canon_sha256":"eea68f4836315791cc19dae157eb0bd8090d4eda3bc595d3b289e0f01ce802ce","abstract_canon_sha256":"051b183f83c49eed3438e43b03899ad16aedd37ca716d9fef9fae4015747b6e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:16:58.814032Z","signature_b64":"D2USjS24Ox1XOWCH11YgfYYBVdKqgtYi02N/vj58Cj1AhQ4t0HKwtybcCW3HPO2Zz8gZspg4EoYhksmVMJz5Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"601a43282d691eb8d40057edef6c75b260452428237f0822a14e24f001c693d0","last_reissued_at":"2026-07-05T01:16:58.813622Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:16:58.813622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the classification of solutions to a weighted elliptic system involving the Grushin operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Foued Mtiri","submitted_at":"2020-07-06T18:53:04Z","abstract_excerpt":"We investigate here the following weighted degenerate elliptic system \\begin{align*} -\\Delta_{s} u = \\Big(1+\\|\\mathbf{x}\\|^{2(s+1)}\\Big)^{\\frac{\\alpha}{2(s+1)}} v^p, \\quad -\\Delta_{s} v = \\Big(1+\\|\\mathbf{x}\\|^{2(s+1)}\\Big)^{\\frac{\\alpha}{2(s+1)}}u^\\theta, \\quad u,v>0\\quad\\mbox{in }\\; \\mathbb{R}^N:=\\mathbb{R}^{N_1}\\times \\mathbb{R}^{N_2}. \\end{align*} where $\\Delta_{s}=\\Delta_{x}+|x|^{2s}\\Delta_{y},$ is the Grushin operator, $s \\geq 0,$ $\\alpha \\geq 0$ and $1<p\\leq\\theta.$ Here $$\\|\\mathbf{x}\\|=\\Big(|x|^{2(s+1)}+|y|^2\\Big)^{\\frac{1}{2(s+1)}}, \\;\\mbox{and}\\;\\; \\mathbf{x}:=(x, y)\\in \\mathbb{R}^N"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.03009","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/2007.03009/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":"2007.03009","created_at":"2026-07-05T01:16:58.813678+00:00"},{"alias_kind":"arxiv_version","alias_value":"2007.03009v1","created_at":"2026-07-05T01:16:58.813678+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.03009","created_at":"2026-07-05T01:16:58.813678+00:00"},{"alias_kind":"pith_short_12","alias_value":"MANEGKBNNEPL","created_at":"2026-07-05T01:16:58.813678+00:00"},{"alias_kind":"pith_short_16","alias_value":"MANEGKBNNEPLRVAA","created_at":"2026-07-05T01:16:58.813678+00:00"},{"alias_kind":"pith_short_8","alias_value":"MANEGKBN","created_at":"2026-07-05T01:16:58.813678+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/MANEGKBNNEPLRVAAK7W663DVWJ","json":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ.json","graph_json":"https://pith.science/api/pith-number/MANEGKBNNEPLRVAAK7W663DVWJ/graph.json","events_json":"https://pith.science/api/pith-number/MANEGKBNNEPLRVAAK7W663DVWJ/events.json","paper":"https://pith.science/paper/MANEGKBN"},"agent_actions":{"view_html":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ","download_json":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ.json","view_paper":"https://pith.science/paper/MANEGKBN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2007.03009&json=true","fetch_graph":"https://pith.science/api/pith-number/MANEGKBNNEPLRVAAK7W663DVWJ/graph.json","fetch_events":"https://pith.science/api/pith-number/MANEGKBNNEPLRVAAK7W663DVWJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ/action/storage_attestation","attest_author":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ/action/author_attestation","sign_citation":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ/action/citation_signature","submit_replication":"https://pith.science/pith/MANEGKBNNEPLRVAAK7W663DVWJ/action/replication_record"}},"created_at":"2026-07-05T01:16:58.813678+00:00","updated_at":"2026-07-05T01:16:58.813678+00:00"}