{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VDG732T42ERDNYT3PUSGC4GN5V","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":"18957045ef1f5789886e841da0ab2352f43db9bf0f827a9783991f1f5bfbb7f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-26T08:04:37Z","title_canon_sha256":"e21d67f693f4b60d8164ae3c346ac35fe6b01ae1d8fab60f49d6bbd414eda2d2"},"schema_version":"1.0","source":{"id":"1607.07580","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07580","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07580v1","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07580","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"pith_short_12","alias_value":"VDG732T42ERD","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VDG732T42ERDNYT3","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VDG732T4","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:2a216414e2eca3529bdc80136b419004b861959a0e814ab58f036485c29cce5d","target":"graph","created_at":"2026-05-18T01:10:28Z","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 article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \\begin{align*} (-\\Delta)_p^{\\alpha}u - \\mu \\frac{|u|^{p-2}u}{|x|^{p\\alpha}} = \\lambda V(x) |u|^{p-2}u \\; \\text{in}\\; \\Omega, \\quad u = 0 \\; \\mbox{in}\\; \\mathbb{R}^n \\setminus\\Omega, \\end{align*} where $n>p\\alpha$, $p\\geq2$, $\\alpha\\in(0,1)$, $0\\leq \\mu <C_{n,\\alpha,p}$ and $\\Omega$ is a domain in $\\mathbb{R}^n$ with Lipschitz boundary containing $0$. In particular, $\\Omega=\\mathbb{R}^n$ is admitted. The weight function $V$ may change sign and may have singular points. We also show that the least positive eigenv","authors_text":"Sarika Goyal","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-26T08:04:37Z","title":"A note on the eigenvalues of fractional Hardy-Sobolev operator with indefinite weight"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07580","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":"sha256:0bbceba61f403aa794eb2542647072b18a5dd157cdb2a77bce19f8c7a7145488","target":"record","created_at":"2026-05-18T01:10:28Z","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":"18957045ef1f5789886e841da0ab2352f43db9bf0f827a9783991f1f5bfbb7f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-26T08:04:37Z","title_canon_sha256":"e21d67f693f4b60d8164ae3c346ac35fe6b01ae1d8fab60f49d6bbd414eda2d2"},"schema_version":"1.0","source":{"id":"1607.07580","kind":"arxiv","version":1}},"canonical_sha256":"a8cdfdea7cd12236e27b7d246170cded6928d479ed46e1d63efaecf6bc26cd66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8cdfdea7cd12236e27b7d246170cded6928d479ed46e1d63efaecf6bc26cd66","first_computed_at":"2026-05-18T01:10:28.239979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:28.239979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+90rqnFtWnOkBUzYnh5uFZd4leok1/IIKiRuGYm4TE8iewjHJKuH9X+YtLMqu4p32kA2XB6+6C1T5Pgxwq/nBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:28.240506Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.07580","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0bbceba61f403aa794eb2542647072b18a5dd157cdb2a77bce19f8c7a7145488","sha256:2a216414e2eca3529bdc80136b419004b861959a0e814ab58f036485c29cce5d"],"state_sha256":"001bb231dee8f1a62a473e6cc1fe0d06bcf9e5fe87e49f3a3b7578fad1c48e60"}