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Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $\\alpha$, $p$ and $\\lambda$."},"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":"1511.09124","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-30T01:05:07Z","cross_cats_sorted":[],"title_canon_sha256":"dbf288c2fd846c1a2232f4e1da9dd87634c78518ae53ac76f41a5b2cb2869c6a","abstract_canon_sha256":"d5c0e59ca104ba26817687aa0e27f55e44e48a4de21b02d55b394717b1e224a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:25.922562Z","signature_b64":"OOGRyo91dhjhnD6iikNICBoUP72sCjOjM+UrqbtGkArwrRNCpGa8eZTF9OMTo48nnEUXTNqWcHpql7kMheIpDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91c1109bd6130470614bb7aa580dc4d0621ea080d01bfb841fe5fd653e95ac02","last_reissued_at":"2026-05-18T01:25:25.921986Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:25.921986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional nonlinear Schr\\\"odinger equations with singular potential in $\\mathbf R^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guoyuan Chen, Youquan Zheng","submitted_at":"2015-11-30T01:05:07Z","abstract_excerpt":"We are interested in nonlinear fractional Schr\\\"odinger equations with singular potential of form \\begin{equation*} (-\\Delta)^su=\\frac{\\lambda}{|x|^{\\alpha}}u+|u|^{p-1}u,\\quad \\mathbf R^n\\setminus\\{0\\}, \\end{equation*} where $s\\in (0,1)$, $\\alpha>0$, $p\\ge1$ and $\\lambda\\in \\mathbf R$. Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $\\alpha$, $p$ and $\\lambda$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09124","kind":"arxiv","version":2},"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":"1511.09124","created_at":"2026-05-18T01:25:25.922070+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.09124v2","created_at":"2026-05-18T01:25:25.922070+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09124","created_at":"2026-05-18T01:25:25.922070+00:00"},{"alias_kind":"pith_short_12","alias_value":"SHARBG6WCMCH","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"SHARBG6WCMCHAYKL","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"SHARBG6W","created_at":"2026-05-18T12:29:42.218222+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/SHARBG6WCMCHAYKLW6VFQDOE2B","json":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B.json","graph_json":"https://pith.science/api/pith-number/SHARBG6WCMCHAYKLW6VFQDOE2B/graph.json","events_json":"https://pith.science/api/pith-number/SHARBG6WCMCHAYKLW6VFQDOE2B/events.json","paper":"https://pith.science/paper/SHARBG6W"},"agent_actions":{"view_html":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B","download_json":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B.json","view_paper":"https://pith.science/paper/SHARBG6W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.09124&json=true","fetch_graph":"https://pith.science/api/pith-number/SHARBG6WCMCHAYKLW6VFQDOE2B/graph.json","fetch_events":"https://pith.science/api/pith-number/SHARBG6WCMCHAYKLW6VFQDOE2B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B/action/storage_attestation","attest_author":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B/action/author_attestation","sign_citation":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B/action/citation_signature","submit_replication":"https://pith.science/pith/SHARBG6WCMCHAYKLW6VFQDOE2B/action/replication_record"}},"created_at":"2026-05-18T01:25:25.922070+00:00","updated_at":"2026-05-18T01:25:25.922070+00:00"}