{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:WEV4ZHGF6ANH3QIQSJXSF6LKRH","short_pith_number":"pith:WEV4ZHGF","schema_version":"1.0","canonical_sha256":"b12bcc9cc5f01a7dc110926f22f96a89eaf8f25bec3720599d54cb26e223059b","source":{"kind":"arxiv","id":"1707.07820","version":1},"attestation_state":"computed","paper":{"title":"Nonlinear Choquard equations: doubly critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinmyoung Seok","submitted_at":"2017-07-25T06:02:46Z","abstract_excerpt":"Consider nonlinear Choquard equations \\begin{equation*} \\left\\{\\begin{array}{rcl} -\\Delta u +u & = &(I_\\alpha*F(u))F'(u) \\quad \\text{in } \\mathbb{R}^N, \\\\ \\lim_{x \\to \\infty}u(x) & = &0, \\end{array}\\right. \\end{equation*} where $I_\\alpha$ denotes Riesz potential and $\\alpha \\in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) = \\frac{N}{N+\\alpha}|u|^{\\frac{N+\\alpha}{N}}+\\frac{N-2}{N+\\alpha}|u|^{\\frac{N+\\alpha}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N \\geq 5$ and $\\alpha + 4 < N$."},"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":"1707.07820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-25T06:02:46Z","cross_cats_sorted":[],"title_canon_sha256":"b4d8599b47e496c1d52a9a5f1549260835c6e13dc6f89db07020ea572897f817","abstract_canon_sha256":"36505a403a11f4bb181d9c6d88e0748529e600ebc17213885d16efc551665268"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:30.210798Z","signature_b64":"A+NOlYRH8jbgrjKM2ZqNCs9pi/xIyUUpk2C6n75UsIlGIOpnQpYbQykd/gTZpzr8tG6qWH/mqy9yttuYgbaCBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b12bcc9cc5f01a7dc110926f22f96a89eaf8f25bec3720599d54cb26e223059b","last_reissued_at":"2026-05-18T00:39:30.210144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:30.210144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonlinear Choquard equations: doubly critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinmyoung Seok","submitted_at":"2017-07-25T06:02:46Z","abstract_excerpt":"Consider nonlinear Choquard equations \\begin{equation*} \\left\\{\\begin{array}{rcl} -\\Delta u +u & = &(I_\\alpha*F(u))F'(u) \\quad \\text{in } \\mathbb{R}^N, \\\\ \\lim_{x \\to \\infty}u(x) & = &0, \\end{array}\\right. \\end{equation*} where $I_\\alpha$ denotes Riesz potential and $\\alpha \\in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) = \\frac{N}{N+\\alpha}|u|^{\\frac{N+\\alpha}{N}}+\\frac{N-2}{N+\\alpha}|u|^{\\frac{N+\\alpha}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N \\geq 5$ and $\\alpha + 4 < N$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07820","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":"1707.07820","created_at":"2026-05-18T00:39:30.210236+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.07820v1","created_at":"2026-05-18T00:39:30.210236+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.07820","created_at":"2026-05-18T00:39:30.210236+00:00"},{"alias_kind":"pith_short_12","alias_value":"WEV4ZHGF6ANH","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WEV4ZHGF6ANH3QIQ","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WEV4ZHGF","created_at":"2026-05-18T12:31:53.515858+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/WEV4ZHGF6ANH3QIQSJXSF6LKRH","json":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH.json","graph_json":"https://pith.science/api/pith-number/WEV4ZHGF6ANH3QIQSJXSF6LKRH/graph.json","events_json":"https://pith.science/api/pith-number/WEV4ZHGF6ANH3QIQSJXSF6LKRH/events.json","paper":"https://pith.science/paper/WEV4ZHGF"},"agent_actions":{"view_html":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH","download_json":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH.json","view_paper":"https://pith.science/paper/WEV4ZHGF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.07820&json=true","fetch_graph":"https://pith.science/api/pith-number/WEV4ZHGF6ANH3QIQSJXSF6LKRH/graph.json","fetch_events":"https://pith.science/api/pith-number/WEV4ZHGF6ANH3QIQSJXSF6LKRH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH/action/storage_attestation","attest_author":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH/action/author_attestation","sign_citation":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH/action/citation_signature","submit_replication":"https://pith.science/pith/WEV4ZHGF6ANH3QIQSJXSF6LKRH/action/replication_record"}},"created_at":"2026-05-18T00:39:30.210236+00:00","updated_at":"2026-05-18T00:39:30.210236+00:00"}