{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:GGNDUFEKLAZCI6YO4AYBSFBLT4","short_pith_number":"pith:GGNDUFEK","schema_version":"1.0","canonical_sha256":"319a3a148a5832247b0ee03019142b9f3d3002ee1db4481288ead161b431e926","source":{"kind":"arxiv","id":"1508.06006","version":1},"attestation_state":"computed","paper":{"title":"On critical $p$-Laplacian systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kanishka Perera, Wenming Zou, Zhenyu Guo","submitted_at":"2015-08-25T01:49:35Z","abstract_excerpt":"We consider the critical $p$-Laplacian system \\begin{equation}\\label{92} \\begin{cases}-\\Delta_p u-\\frac{\\lambda a}{p}|u|^{a-2}u|v|^b =\\mu_1|u|^{p^\\ast-2}u+\\frac{\\alpha\\gamma}{p^\\ast}|u|^{\\alpha-2}u|v|^{\\beta}, &x\\in\\Omega,\\\\ -\\Delta_p v-\\frac{\\lambda b}{p}|u|^a|v|^{b-2}v =\\mu_2|v|^{p^\\ast-2}v+\\frac{\\beta\\gamma}{p^\\ast}|u|^{\\alpha}|v|^{\\beta-2}v, &x\\in\\Omega,\\\\ u,v\\ \\text{in } D_0^{1,p}(\\Omega), \\end{cases} \\end{equation} where $\\Delta_p:=\\text{div}(|\\nabla u|^{p-2}\\nabla u)$ is the $p$-Laplacian operator defined on $D^{1,p}(\\mathbb{R}^N):=\\{u\\in L^{p^\\ast}(\\mathbb{R}^N):|\\nabla u|\\in L^p(\\math"},"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":"1508.06006","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-25T01:49:35Z","cross_cats_sorted":[],"title_canon_sha256":"25ae58a878ff79fd49218ef626b148f3905c4fd3c1e33a0a0be3fb8ebb1df67d","abstract_canon_sha256":"f6a28edfa8ee59a991f5636bde016b02ff929191cf71f2ef3a0e294398095e1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:47.158305Z","signature_b64":"At3bHaYOGIhgyBYke+TyR5ygQulGYF8DZyjLbFHUDSk4nEToYNTKFIsR8ErB9wjeKNR6TtcuZrtMfNfOQOZ/DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"319a3a148a5832247b0ee03019142b9f3d3002ee1db4481288ead161b431e926","last_reissued_at":"2026-05-18T01:34:47.157711Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:47.157711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On critical $p$-Laplacian systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kanishka Perera, Wenming Zou, Zhenyu Guo","submitted_at":"2015-08-25T01:49:35Z","abstract_excerpt":"We consider the critical $p$-Laplacian system \\begin{equation}\\label{92} \\begin{cases}-\\Delta_p u-\\frac{\\lambda a}{p}|u|^{a-2}u|v|^b =\\mu_1|u|^{p^\\ast-2}u+\\frac{\\alpha\\gamma}{p^\\ast}|u|^{\\alpha-2}u|v|^{\\beta}, &x\\in\\Omega,\\\\ -\\Delta_p v-\\frac{\\lambda b}{p}|u|^a|v|^{b-2}v =\\mu_2|v|^{p^\\ast-2}v+\\frac{\\beta\\gamma}{p^\\ast}|u|^{\\alpha}|v|^{\\beta-2}v, &x\\in\\Omega,\\\\ u,v\\ \\text{in } D_0^{1,p}(\\Omega), \\end{cases} \\end{equation} where $\\Delta_p:=\\text{div}(|\\nabla u|^{p-2}\\nabla u)$ is the $p$-Laplacian operator defined on $D^{1,p}(\\mathbb{R}^N):=\\{u\\in L^{p^\\ast}(\\mathbb{R}^N):|\\nabla u|\\in L^p(\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06006","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":"1508.06006","created_at":"2026-05-18T01:34:47.157818+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.06006v1","created_at":"2026-05-18T01:34:47.157818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06006","created_at":"2026-05-18T01:34:47.157818+00:00"},{"alias_kind":"pith_short_12","alias_value":"GGNDUFEKLAZC","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"GGNDUFEKLAZCI6YO","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"GGNDUFEK","created_at":"2026-05-18T12:29:22.688609+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/GGNDUFEKLAZCI6YO4AYBSFBLT4","json":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4.json","graph_json":"https://pith.science/api/pith-number/GGNDUFEKLAZCI6YO4AYBSFBLT4/graph.json","events_json":"https://pith.science/api/pith-number/GGNDUFEKLAZCI6YO4AYBSFBLT4/events.json","paper":"https://pith.science/paper/GGNDUFEK"},"agent_actions":{"view_html":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4","download_json":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4.json","view_paper":"https://pith.science/paper/GGNDUFEK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.06006&json=true","fetch_graph":"https://pith.science/api/pith-number/GGNDUFEKLAZCI6YO4AYBSFBLT4/graph.json","fetch_events":"https://pith.science/api/pith-number/GGNDUFEKLAZCI6YO4AYBSFBLT4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4/action/storage_attestation","attest_author":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4/action/author_attestation","sign_citation":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4/action/citation_signature","submit_replication":"https://pith.science/pith/GGNDUFEKLAZCI6YO4AYBSFBLT4/action/replication_record"}},"created_at":"2026-05-18T01:34:47.157818+00:00","updated_at":"2026-05-18T01:34:47.157818+00:00"}