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Lin-Ni-Wei \\cite{LNW} proved that there exists $\\ve_0$ such that for $0<\\ve<\\ve_0$ and for each integer $k$ bounded by {equation} 1\\leq k\\leq \\frac{\\delta(\\Omega,n,p)}{(\\ve |\\log \\ve |)^n} {equation} where $\\delta(\\Omega,n,p)$ is a"},"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":"1209.2824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-13T09:13:45Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"dea81b7b20a6a543104e447f42f9453c253113b23c95ad076e4abd3b44e9fc17","abstract_canon_sha256":"f339689fa315201742eba96439f0ea5057c06aa2c258415c54671f4f49710399"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:40.859207Z","signature_b64":"FH/BZV0xfNXA9xiziNf9KTs/AuBudivXq1fHWv6xyG7+wJW6a1Zqh5rOcWlKODbl/+LFa/76swInywwGdreGCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea735f44576a28c84e079b0c7f2b01c8250e26c652d4ed88277277d2708a5826","last_reissued_at":"2026-05-18T03:45:40.858600Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:40.858600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An optimal bound on the number of interior spike solutions for Lin-Ni-Takagi problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Jing Zeng, Juncheng Wei, Weiwei Ao","submitted_at":"2012-09-13T09:13:45Z","abstract_excerpt":"We consider the following singularly perturbed Neumann problem {eqnarray*} \\ve^2 \\Delta u -u +u^p = 0 \\quad {{in}} \\quad \\Omega, \\quad u>0 \\quad {{in}} \\quad \\Omega, \\quad {\\partial u \\over \\partial \\nu}=0 \\quad {{on}} \\quad \\partial \\Omega, {eqnarray*} where $p$ is subcritical and $\\Omega$ is a smooth and bounded domain in $\\R^n$ with its unit outward normal $\\nu$. Lin-Ni-Wei \\cite{LNW} proved that there exists $\\ve_0$ such that for $0<\\ve<\\ve_0$ and for each integer $k$ bounded by {equation} 1\\leq k\\leq \\frac{\\delta(\\Omega,n,p)}{(\\ve |\\log \\ve |)^n} {equation} where $\\delta(\\Omega,n,p)$ is a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2824","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":"1209.2824","created_at":"2026-05-18T03:45:40.858689+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.2824v1","created_at":"2026-05-18T03:45:40.858689+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2824","created_at":"2026-05-18T03:45:40.858689+00:00"},{"alias_kind":"pith_short_12","alias_value":"5JZV6RCXNIUM","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"5JZV6RCXNIUMQTQH","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"5JZV6RCX","created_at":"2026-05-18T12:26:53.410803+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/5JZV6RCXNIUMQTQHTMGH6KYBZA","json":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA.json","graph_json":"https://pith.science/api/pith-number/5JZV6RCXNIUMQTQHTMGH6KYBZA/graph.json","events_json":"https://pith.science/api/pith-number/5JZV6RCXNIUMQTQHTMGH6KYBZA/events.json","paper":"https://pith.science/paper/5JZV6RCX"},"agent_actions":{"view_html":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA","download_json":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA.json","view_paper":"https://pith.science/paper/5JZV6RCX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.2824&json=true","fetch_graph":"https://pith.science/api/pith-number/5JZV6RCXNIUMQTQHTMGH6KYBZA/graph.json","fetch_events":"https://pith.science/api/pith-number/5JZV6RCXNIUMQTQHTMGH6KYBZA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA/action/storage_attestation","attest_author":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA/action/author_attestation","sign_citation":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA/action/citation_signature","submit_replication":"https://pith.science/pith/5JZV6RCXNIUMQTQHTMGH6KYBZA/action/replication_record"}},"created_at":"2026-05-18T03:45:40.858689+00:00","updated_at":"2026-05-18T03:45:40.858689+00:00"}