{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:PGVA2TJFUNBWZFUWR7AYDXT64L","short_pith_number":"pith:PGVA2TJF","schema_version":"1.0","canonical_sha256":"79aa0d4d25a3436c96968fc181de7ee2f80dab725edd19d601b155d766e157d2","source":{"kind":"arxiv","id":"1906.10730","version":1},"attestation_state":"computed","paper":{"title":"The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David G. Costa, Joao Marcos do \\'O, Pawan Kumar Mishra","submitted_at":"2019-06-25T19:14:56Z","abstract_excerpt":"In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth\n  \\begin{align*}\n  L(u)&-\\lambda h(x)|x|^{-2(1+a)}u=\\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\\;\\; \\text{in } \\mathbb R^N,\n  \\end{align*}\n  where\n  $h(x)\\geq 0$, $f(x)$ is a continuous function which may change sign, $\\lambda, \\mu$ are positive real parameters and $1<q<2$, $4< p=2N/[N+2(b-a)-2]$, $0\\leq a<b<a+1<N/2$, $N\\geq 3$. Here\n  $$\n  L(u)=-M\\left(\\int_{\\mathbb R^N} |x|^{-2a}|\\nabla u|^2dx\\right)\\mathrm {div}(|x|^{-2a}\\nabla u)\n  $$\n  and the function $M:\\mathbb R^+\\cup \\"},"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":"1906.10730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-25T19:14:56Z","cross_cats_sorted":[],"title_canon_sha256":"d35cde9b29d1cd962a3fa3d2a0afed45cce24336ea8d910015a62d4821a0da87","abstract_canon_sha256":"9889fc55b4dc0b5851ed070be0c6c4344ea27709889079a481cea7ba38a84e2c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:14.908992Z","signature_b64":"s5BXVngqLpZRkhgSR9aejT5s/Bpaq3fc9Mth2yQPPIDRl2bHNqFe6tFItZjA7aASRTS8xiDNaYkw1NIum5j/AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"79aa0d4d25a3436c96968fc181de7ee2f80dab725edd19d601b155d766e157d2","last_reissued_at":"2026-05-17T23:42:14.908181Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:14.908181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David G. Costa, Joao Marcos do \\'O, Pawan Kumar Mishra","submitted_at":"2019-06-25T19:14:56Z","abstract_excerpt":"In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth\n  \\begin{align*}\n  L(u)&-\\lambda h(x)|x|^{-2(1+a)}u=\\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\\;\\; \\text{in } \\mathbb R^N,\n  \\end{align*}\n  where\n  $h(x)\\geq 0$, $f(x)$ is a continuous function which may change sign, $\\lambda, \\mu$ are positive real parameters and $1<q<2$, $4< p=2N/[N+2(b-a)-2]$, $0\\leq a<b<a+1<N/2$, $N\\geq 3$. Here\n  $$\n  L(u)=-M\\left(\\int_{\\mathbb R^N} |x|^{-2a}|\\nabla u|^2dx\\right)\\mathrm {div}(|x|^{-2a}\\nabla u)\n  $$\n  and the function $M:\\mathbb R^+\\cup \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.10730","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":"1906.10730","created_at":"2026-05-17T23:42:14.908331+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.10730v1","created_at":"2026-05-17T23:42:14.908331+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.10730","created_at":"2026-05-17T23:42:14.908331+00:00"},{"alias_kind":"pith_short_12","alias_value":"PGVA2TJFUNBW","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"PGVA2TJFUNBWZFUW","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"PGVA2TJF","created_at":"2026-05-18T12:33:24.271573+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/PGVA2TJFUNBWZFUWR7AYDXT64L","json":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L.json","graph_json":"https://pith.science/api/pith-number/PGVA2TJFUNBWZFUWR7AYDXT64L/graph.json","events_json":"https://pith.science/api/pith-number/PGVA2TJFUNBWZFUWR7AYDXT64L/events.json","paper":"https://pith.science/paper/PGVA2TJF"},"agent_actions":{"view_html":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L","download_json":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L.json","view_paper":"https://pith.science/paper/PGVA2TJF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.10730&json=true","fetch_graph":"https://pith.science/api/pith-number/PGVA2TJFUNBWZFUWR7AYDXT64L/graph.json","fetch_events":"https://pith.science/api/pith-number/PGVA2TJFUNBWZFUWR7AYDXT64L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L/action/storage_attestation","attest_author":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L/action/author_attestation","sign_citation":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L/action/citation_signature","submit_replication":"https://pith.science/pith/PGVA2TJFUNBWZFUWR7AYDXT64L/action/replication_record"}},"created_at":"2026-05-17T23:42:14.908331+00:00","updated_at":"2026-05-17T23:42:14.908331+00:00"}