{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:PGVA2TJFUNBWZFUWR7AYDXT64L","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9889fc55b4dc0b5851ed070be0c6c4344ea27709889079a481cea7ba38a84e2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-25T19:14:56Z","title_canon_sha256":"d35cde9b29d1cd962a3fa3d2a0afed45cce24336ea8d910015a62d4821a0da87"},"schema_version":"1.0","source":{"id":"1906.10730","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.10730","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"arxiv_version","alias_value":"1906.10730v1","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.10730","created_at":"2026-05-17T23:42:14Z"},{"alias_kind":"pith_short_12","alias_value":"PGVA2TJFUNBW","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"PGVA2TJFUNBWZFUW","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"PGVA2TJF","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:3ebe807815d73aacba226a3bb0af068f5f69d81dc8518fd0776361eb9cb805c0","target":"graph","created_at":"2026-05-17T23:42:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 \\","authors_text":"David G. Costa, Joao Marcos do \\'O, Pawan Kumar Mishra","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-25T19:14:56Z","title":"The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.10730","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:08be435f0e8549715050ed7be17d808c5887ea987362f5579ca06980d02ea343","target":"record","created_at":"2026-05-17T23:42:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9889fc55b4dc0b5851ed070be0c6c4344ea27709889079a481cea7ba38a84e2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-25T19:14:56Z","title_canon_sha256":"d35cde9b29d1cd962a3fa3d2a0afed45cce24336ea8d910015a62d4821a0da87"},"schema_version":"1.0","source":{"id":"1906.10730","kind":"arxiv","version":1}},"canonical_sha256":"79aa0d4d25a3436c96968fc181de7ee2f80dab725edd19d601b155d766e157d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"79aa0d4d25a3436c96968fc181de7ee2f80dab725edd19d601b155d766e157d2","first_computed_at":"2026-05-17T23:42:14.908181Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:14.908181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s5BXVngqLpZRkhgSR9aejT5s/Bpaq3fc9Mth2yQPPIDRl2bHNqFe6tFItZjA7aASRTS8xiDNaYkw1NIum5j/AA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:14.908992Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.10730","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:08be435f0e8549715050ed7be17d808c5887ea987362f5579ca06980d02ea343","sha256:3ebe807815d73aacba226a3bb0af068f5f69d81dc8518fd0776361eb9cb805c0"],"state_sha256":"49023d178072cdbf55d36361e7d34f3cb31e1f0b419ab6b29a4dba3a4c1a3fe5"}