{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:LMWFI3IY2FOLQVJ4G7EBO6I4QG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fea1a777aa81717522a108eaebf4c56dda752179e11314f324f250ef6dfbb21b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-21T06:28:35Z","title_canon_sha256":"c31bf7747e31783a60b49691d4bbcac7f45cb16ae6ec73dbc16e26bebf2fc2f4"},"schema_version":"1.0","source":{"id":"1901.06800","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.06800","created_at":"2026-05-17T23:55:49Z"},{"alias_kind":"arxiv_version","alias_value":"1901.06800v1","created_at":"2026-05-17T23:55:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.06800","created_at":"2026-05-17T23:55:49Z"},{"alias_kind":"pith_short_12","alias_value":"LMWFI3IY2FOL","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"LMWFI3IY2FOLQVJ4","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"LMWFI3IY","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:84cb9a42c612d794356a8a7f6a730e17c1ac0615dc5040bbb62d187b555a36f1","target":"graph","created_at":"2026-05-17T23:55:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we study the following conformally invariant poly-harmonic equation $$\\Delta^mu=-u^\\frac{3+2m}{3-2m}\\quad\\text{in }\\mathbb{R}^3,\\quad u>0,$$ with $m=2,3$. We prove the existence of positive smooth radial solutions with prescribed volume $\\int_{\\mathbb{R}^3} u^\\frac{6}{3-2m}dx$. We show that the set of all possible values of the volume is a bounded interval $(0,\\Lambda^*]$ for $m=2$, and it is $(0,\\infty)$ for $m=3$. This is in sharp contrast to $m=1$ case in which the volume $\\int_{\\mathbb{R}^3} u^\\frac{6}{3-2m}dx$ is a fixed value.","authors_text":"Ali Hyder, Juncheng Wei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-21T06:28:35Z","title":"Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.06800","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:84049721f389c0afb4d0c8216daa97a9cc39cfc37801c5183ec87652bd211094","target":"record","created_at":"2026-05-17T23:55:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fea1a777aa81717522a108eaebf4c56dda752179e11314f324f250ef6dfbb21b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-21T06:28:35Z","title_canon_sha256":"c31bf7747e31783a60b49691d4bbcac7f45cb16ae6ec73dbc16e26bebf2fc2f4"},"schema_version":"1.0","source":{"id":"1901.06800","kind":"arxiv","version":1}},"canonical_sha256":"5b2c546d18d15cb8553c37c817791c81a4ef40c17dddeef691feb31665265e24","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b2c546d18d15cb8553c37c817791c81a4ef40c17dddeef691feb31665265e24","first_computed_at":"2026-05-17T23:55:49.109740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:49.109740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XV+9SHifGS6oJWqmkZ/djULUJygKgBYJNYq//sLcIBGT3EKNtcwRAFV3T7VI8hBCQE+EWmiOYbh1ROkbjkB9Cw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:49.110299Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.06800","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84049721f389c0afb4d0c8216daa97a9cc39cfc37801c5183ec87652bd211094","sha256:84cb9a42c612d794356a8a7f6a730e17c1ac0615dc5040bbb62d187b555a36f1"],"state_sha256":"25c1dde4e160013c34497610b15ae8076bfe5ca50fcffb89eb6858be8c057fdd"}