{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:N3HSQUWDGLW5Z4YDB25JLMINHH","short_pith_number":"pith:N3HSQUWD","schema_version":"1.0","canonical_sha256":"6ecf2852c332eddcf3030eba95b10d39c71d5ca23613d4d30f93c1aea58094fa","source":{"kind":"arxiv","id":"1712.08598","version":2},"attestation_state":"computed","paper":{"title":"Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tom\\'as Sanz-Perela","submitted_at":"2017-12-22T17:55:36Z","abstract_excerpt":"We study the regularity of stable solutions to the problem $$ \\left\\{ \\begin{array}{rcll} (-\\Delta)^s u &=& f(u) & \\text{in} \\quad B_1\\,, u &\\equiv&0 & \\text{in} \\quad \\mathbb R^n\\setminus B_1\\,, \\end{array} \\right. $$ where $s\\in(0,1)$. Our main result establishes an $L^\\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \\leq n < 2(s+2+\\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\\in(0,1)$ in dimensions $2 \\leq n\\leq 6$. It applies to all nonlinearities $f\\in C^2$.\n  For such parameters $s$ and $n$, our result leads to the regular"},"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":"1712.08598","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-22T17:55:36Z","cross_cats_sorted":[],"title_canon_sha256":"f573ac87cc541589d796c02b541ccb0ffdbeb9b3aa8e2a4aadcfb98276f6e7d9","abstract_canon_sha256":"e5a31abd87673d65c8e6bec20f857ff94f0d4d9e58512d1de91a3f6e65c472b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:28.106675Z","signature_b64":"BrsUaD28R1UOuAsgBYMQUhWElAlat8AuNO6p+t3dtYPymovqHrsjCgeyE8cA4fu3PwiGnTnlHmtiYxKLFmBcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ecf2852c332eddcf3030eba95b10d39c71d5ca23613d4d30f93c1aea58094fa","last_reissued_at":"2026-05-18T00:11:28.106298Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:28.106298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tom\\'as Sanz-Perela","submitted_at":"2017-12-22T17:55:36Z","abstract_excerpt":"We study the regularity of stable solutions to the problem $$ \\left\\{ \\begin{array}{rcll} (-\\Delta)^s u &=& f(u) & \\text{in} \\quad B_1\\,, u &\\equiv&0 & \\text{in} \\quad \\mathbb R^n\\setminus B_1\\,, \\end{array} \\right. $$ where $s\\in(0,1)$. Our main result establishes an $L^\\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \\leq n < 2(s+2+\\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\\in(0,1)$ in dimensions $2 \\leq n\\leq 6$. It applies to all nonlinearities $f\\in C^2$.\n  For such parameters $s$ and $n$, our result leads to the regular"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08598","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":"1712.08598","created_at":"2026-05-18T00:11:28.106369+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.08598v2","created_at":"2026-05-18T00:11:28.106369+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08598","created_at":"2026-05-18T00:11:28.106369+00:00"},{"alias_kind":"pith_short_12","alias_value":"N3HSQUWDGLW5","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"N3HSQUWDGLW5Z4YD","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"N3HSQUWD","created_at":"2026-05-18T12:31:31.346846+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/N3HSQUWDGLW5Z4YDB25JLMINHH","json":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH.json","graph_json":"https://pith.science/api/pith-number/N3HSQUWDGLW5Z4YDB25JLMINHH/graph.json","events_json":"https://pith.science/api/pith-number/N3HSQUWDGLW5Z4YDB25JLMINHH/events.json","paper":"https://pith.science/paper/N3HSQUWD"},"agent_actions":{"view_html":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH","download_json":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH.json","view_paper":"https://pith.science/paper/N3HSQUWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.08598&json=true","fetch_graph":"https://pith.science/api/pith-number/N3HSQUWDGLW5Z4YDB25JLMINHH/graph.json","fetch_events":"https://pith.science/api/pith-number/N3HSQUWDGLW5Z4YDB25JLMINHH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH/action/storage_attestation","attest_author":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH/action/author_attestation","sign_citation":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH/action/citation_signature","submit_replication":"https://pith.science/pith/N3HSQUWDGLW5Z4YDB25JLMINHH/action/replication_record"}},"created_at":"2026-05-18T00:11:28.106369+00:00","updated_at":"2026-05-18T00:11:28.106369+00:00"}