{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:TQ3XONBQWJRDKONMWQWG6EURAI","short_pith_number":"pith:TQ3XONBQ","schema_version":"1.0","canonical_sha256":"9c37773430b2623539acb42c6f12910210fcaef7c8d6db173c1de9ba142abc38","source":{"kind":"arxiv","id":"1305.4426","version":1},"attestation_state":"computed","paper":{"title":"Concentration phenomenon for fractional nonlinear Schr\\\"{o}dinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Guoyuan Chen, Youquan Zheng","submitted_at":"2013-05-20T02:38:30Z","abstract_excerpt":"We study the concentration phenomenon for solutions of the fractional nonlinear Schr\\\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \\begin{equation}\\label{e:abstract} (-\\varepsilon^2\\Delta)^sv+Vv-|v|^{\\alpha}v=0\\quad\\mbox{in}\\quad\\mathbf R^n, \\end{equation} where $n =1, 2, 3$, $\\max\\{\\frac{1}{2}, \\frac{n}{4}\\}< s < 1$, $1 \\leq \\alpha < \\alpha_*(s,n)$, $V\\in C^3_{b}(\\mathbf{R}^n)$. Here the exponent $\\alpha_*(s,n)=\\frac{4s}{n-2s}$ for $0 < s < \\frac{n}{2}$ and $\\alpha_*(s,n)=\\infty$ for $s \\geq\\frac{n}{2}$."},"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":"1305.4426","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-05-20T02:38:30Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"790790ee5603ef94d80413aaf9d0b43ac163fc6b6a939f9d62c1bc0aed04bcb1","abstract_canon_sha256":"07d67e5d5cf2ea6800cf9c8c875743cbce0429546402471f42fdb9d202b3cb69"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:23.753793Z","signature_b64":"bXUgaZnFfAeOZUXoGGORlBtj4Jifu9r/Y1rei5giheWgBova+bIOicwTzBftp0bVVmxfulCCaZr2NRNrb9PNBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c37773430b2623539acb42c6f12910210fcaef7c8d6db173c1de9ba142abc38","last_reissued_at":"2026-05-18T03:25:23.753197Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:23.753197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Concentration phenomenon for fractional nonlinear Schr\\\"{o}dinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Guoyuan Chen, Youquan Zheng","submitted_at":"2013-05-20T02:38:30Z","abstract_excerpt":"We study the concentration phenomenon for solutions of the fractional nonlinear Schr\\\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \\begin{equation}\\label{e:abstract} (-\\varepsilon^2\\Delta)^sv+Vv-|v|^{\\alpha}v=0\\quad\\mbox{in}\\quad\\mathbf R^n, \\end{equation} where $n =1, 2, 3$, $\\max\\{\\frac{1}{2}, \\frac{n}{4}\\}< s < 1$, $1 \\leq \\alpha < \\alpha_*(s,n)$, $V\\in C^3_{b}(\\mathbf{R}^n)$. Here the exponent $\\alpha_*(s,n)=\\frac{4s}{n-2s}$ for $0 < s < \\frac{n}{2}$ and $\\alpha_*(s,n)=\\infty$ for $s \\geq\\frac{n}{2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4426","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":"1305.4426","created_at":"2026-05-18T03:25:23.753299+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.4426v1","created_at":"2026-05-18T03:25:23.753299+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4426","created_at":"2026-05-18T03:25:23.753299+00:00"},{"alias_kind":"pith_short_12","alias_value":"TQ3XONBQWJRD","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"TQ3XONBQWJRDKONM","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"TQ3XONBQ","created_at":"2026-05-18T12:28:02.375192+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/TQ3XONBQWJRDKONMWQWG6EURAI","json":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI.json","graph_json":"https://pith.science/api/pith-number/TQ3XONBQWJRDKONMWQWG6EURAI/graph.json","events_json":"https://pith.science/api/pith-number/TQ3XONBQWJRDKONMWQWG6EURAI/events.json","paper":"https://pith.science/paper/TQ3XONBQ"},"agent_actions":{"view_html":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI","download_json":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI.json","view_paper":"https://pith.science/paper/TQ3XONBQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.4426&json=true","fetch_graph":"https://pith.science/api/pith-number/TQ3XONBQWJRDKONMWQWG6EURAI/graph.json","fetch_events":"https://pith.science/api/pith-number/TQ3XONBQWJRDKONMWQWG6EURAI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI/action/storage_attestation","attest_author":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI/action/author_attestation","sign_citation":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI/action/citation_signature","submit_replication":"https://pith.science/pith/TQ3XONBQWJRDKONMWQWG6EURAI/action/replication_record"}},"created_at":"2026-05-18T03:25:23.753299+00:00","updated_at":"2026-05-18T03:25:23.753299+00:00"}