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For any open domain $\\Omega\\subset\\mathbb{R}^{2m}$, non-positive function $\\varphi\\in C^\\infty(\\Omega)$ such that $\\Delta^m \\varphi\\equiv 0$, and bounded sequence $(V_k)\\subset L^\\infty(\\Omega)$ we prove the existence of a sequence of functions $(u_k)\\subset C^{2m-1}(\\Omega)$ solving the Liouville equation of order $2m$ $$(-\\Delta)^m u_k = V_ke^{2mu_k}\\quad \\text{in }\\Omega, \\quad \\limsup_{k\\to\\infty} \\int_\\Omega e^{2mu_k}dx<\\infty,$$ and blowing up exactly on the set $S_{\\varphi}:=\\{x\\in \\Omega:\\varphi(x)=0\\}$, i.e. $$\\lim_{k\\to\\infty} u_k(x)=+\\infty \\text{ for }x\\"},"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":"1610.06820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-21T15:17:55Z","cross_cats_sorted":[],"title_canon_sha256":"0676270aa193975a2da26412f4d93a5f483c054764aa5fbed22a38582d8afdeb","abstract_canon_sha256":"21d74c07a72d21aeabab3bfdc1bb719f139c86073a8dfc87edefbe3cb26f6da8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:41.628292Z","signature_b64":"2zw+zVZxpg4SCRnZ2cqmf0B8Sjo8afzbsypZtfc+x7OhKptrkjeapcxGCZ72Bw2+3xkWGEsn1AvXenHy3I63Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6809023fc888c5ba153fd687d7a2e5c6e33a960a91e6ad4b81bc9327df6004f","last_reissued_at":"2026-05-18T00:10:41.627484Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:41.627484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder, Luca Martinazzi, Stefano Iula","submitted_at":"2016-10-21T15:17:55Z","abstract_excerpt":"Let $m\\ge 2$ be an integer. For any open domain $\\Omega\\subset\\mathbb{R}^{2m}$, non-positive function $\\varphi\\in C^\\infty(\\Omega)$ such that $\\Delta^m \\varphi\\equiv 0$, and bounded sequence $(V_k)\\subset L^\\infty(\\Omega)$ we prove the existence of a sequence of functions $(u_k)\\subset C^{2m-1}(\\Omega)$ solving the Liouville equation of order $2m$ $$(-\\Delta)^m u_k = V_ke^{2mu_k}\\quad \\text{in }\\Omega, \\quad \\limsup_{k\\to\\infty} \\int_\\Omega e^{2mu_k}dx<\\infty,$$ and blowing up exactly on the set $S_{\\varphi}:=\\{x\\in \\Omega:\\varphi(x)=0\\}$, i.e. $$\\lim_{k\\to\\infty} u_k(x)=+\\infty \\text{ for }x\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06820","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":"1610.06820","created_at":"2026-05-18T00:10:41.627623+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.06820v1","created_at":"2026-05-18T00:10:41.627623+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06820","created_at":"2026-05-18T00:10:41.627623+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y2AJAI74RCGF","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y2AJAI74RCGFXIKT","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y2AJAI74","created_at":"2026-05-18T12:30:51.357362+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/Y2AJAI74RCGFXIKT7VUH26ROLR","json":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR.json","graph_json":"https://pith.science/api/pith-number/Y2AJAI74RCGFXIKT7VUH26ROLR/graph.json","events_json":"https://pith.science/api/pith-number/Y2AJAI74RCGFXIKT7VUH26ROLR/events.json","paper":"https://pith.science/paper/Y2AJAI74"},"agent_actions":{"view_html":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR","download_json":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR.json","view_paper":"https://pith.science/paper/Y2AJAI74","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.06820&json=true","fetch_graph":"https://pith.science/api/pith-number/Y2AJAI74RCGFXIKT7VUH26ROLR/graph.json","fetch_events":"https://pith.science/api/pith-number/Y2AJAI74RCGFXIKT7VUH26ROLR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/action/storage_attestation","attest_author":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/action/author_attestation","sign_citation":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/action/citation_signature","submit_replication":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/action/replication_record"}},"created_at":"2026-05-18T00:10:41.627623+00:00","updated_at":"2026-05-18T00:10:41.627623+00:00"}