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We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when $\\alpha $ is either close to $0$ or close to $N$."},"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":"1606.05668","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-17T20:30:33Z","cross_cats_sorted":[],"title_canon_sha256":"0c91d84e09941dfd57a6aeea49a16df3ab7075c650bca75afacb5e8a136999e9","abstract_canon_sha256":"50eab75a2b0f51ffc97f3619809608c19073b0a800bc283f696c4740e7050916"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:51.119821Z","signature_b64":"yemWIfuNYIjjVSVhan0B2yz+q0DLpmi95baCu96G8z76mDzdiG/4+PkCVv8LGulOQ+oDEsHxqdZvFl+q5bsmDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f89c5d7a1e3f5a54c575daef1105bf767f828c0a2561c88373f55c40a9ace7f","last_reissued_at":"2026-05-18T00:07:51.119140Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:51.119140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Odd symmetry of least energy nodal solutions for the Choquard equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Ruiz, Jean Van Schaftingen","submitted_at":"2016-06-17T20:30:33Z","abstract_excerpt":"We consider the Choquard equation (also known as stationary Hartree equation or Schr\\\"odinger--Newton equation) \\[\n  -\\Delta u + u = (I_\\alpha \\star |u|^p) |u|^{p - 2}u. \\] Here $I_\\alpha$ stands for the Riesz potential of order $\\alpha \\in (0,N)$, and $\\frac{N - 2}{N + \\alpha} < \\frac{1}{p} \\le \\frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when $\\alpha $ is either close to $0$ or close to $N$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05668","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":"1606.05668","created_at":"2026-05-18T00:07:51.119251+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.05668v1","created_at":"2026-05-18T00:07:51.119251+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05668","created_at":"2026-05-18T00:07:51.119251+00:00"},{"alias_kind":"pith_short_12","alias_value":"B6E4LV5B4P22","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"B6E4LV5B4P22KTCX","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"B6E4LV5B","created_at":"2026-05-18T12:30:07.202191+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/B6E4LV5B4P22KTCXLWXPCEC365","json":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365.json","graph_json":"https://pith.science/api/pith-number/B6E4LV5B4P22KTCXLWXPCEC365/graph.json","events_json":"https://pith.science/api/pith-number/B6E4LV5B4P22KTCXLWXPCEC365/events.json","paper":"https://pith.science/paper/B6E4LV5B"},"agent_actions":{"view_html":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365","download_json":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365.json","view_paper":"https://pith.science/paper/B6E4LV5B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.05668&json=true","fetch_graph":"https://pith.science/api/pith-number/B6E4LV5B4P22KTCXLWXPCEC365/graph.json","fetch_events":"https://pith.science/api/pith-number/B6E4LV5B4P22KTCXLWXPCEC365/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365/action/storage_attestation","attest_author":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365/action/author_attestation","sign_citation":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365/action/citation_signature","submit_replication":"https://pith.science/pith/B6E4LV5B4P22KTCXLWXPCEC365/action/replication_record"}},"created_at":"2026-05-18T00:07:51.119251+00:00","updated_at":"2026-05-18T00:07:51.119251+00:00"}