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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."},"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":"1901.06800","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-21T06:28:35Z","cross_cats_sorted":[],"title_canon_sha256":"c31bf7747e31783a60b49691d4bbcac7f45cb16ae6ec73dbc16e26bebf2fc2f4","abstract_canon_sha256":"fea1a777aa81717522a108eaebf4c56dda752179e11314f324f250ef6dfbb21b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:49.110299Z","signature_b64":"XV+9SHifGS6oJWqmkZ/djULUJygKgBYJNYq//sLcIBGT3EKNtcwRAFV3T7VI8hBCQE+EWmiOYbh1ROkbjkB9Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b2c546d18d15cb8553c37c817791c81a4ef40c17dddeef691feb31665265e24","last_reissued_at":"2026-05-17T23:55:49.109740Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:49.109740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder, Juncheng Wei","submitted_at":"2019-01-21T06:28:35Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.06800","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":"1901.06800","created_at":"2026-05-17T23:55:49.109824+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.06800v1","created_at":"2026-05-17T23:55:49.109824+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.06800","created_at":"2026-05-17T23:55:49.109824+00:00"},{"alias_kind":"pith_short_12","alias_value":"LMWFI3IY2FOL","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"LMWFI3IY2FOLQVJ4","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"LMWFI3IY","created_at":"2026-05-18T12:33:21.387695+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/LMWFI3IY2FOLQVJ4G7EBO6I4QG","json":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG.json","graph_json":"https://pith.science/api/pith-number/LMWFI3IY2FOLQVJ4G7EBO6I4QG/graph.json","events_json":"https://pith.science/api/pith-number/LMWFI3IY2FOLQVJ4G7EBO6I4QG/events.json","paper":"https://pith.science/paper/LMWFI3IY"},"agent_actions":{"view_html":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG","download_json":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG.json","view_paper":"https://pith.science/paper/LMWFI3IY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.06800&json=true","fetch_graph":"https://pith.science/api/pith-number/LMWFI3IY2FOLQVJ4G7EBO6I4QG/graph.json","fetch_events":"https://pith.science/api/pith-number/LMWFI3IY2FOLQVJ4G7EBO6I4QG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG/action/storage_attestation","attest_author":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG/action/author_attestation","sign_citation":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG/action/citation_signature","submit_replication":"https://pith.science/pith/LMWFI3IY2FOLQVJ4G7EBO6I4QG/action/replication_record"}},"created_at":"2026-05-17T23:55:49.109824+00:00","updated_at":"2026-05-17T23:55:49.109824+00:00"}