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Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension $n\\geq 3$ we show that to a certain extent the asymptotic behavior of $u$ and the constant $V$ can be prescribed simultaneously. Furthermore if $Q=-(n-1)!$ then $V$ can be chosen to be any positive number. This is in contrast to the case $n=3$, $Q=2$, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily $V\\le |S^"},"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":"1502.02685","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2015-02-09T21:09:50Z","cross_cats_sorted":[],"title_canon_sha256":"44c4833983545ecd2c29b6a68919ab27ad33a079ee655f7699d6c8ffd85248cd","abstract_canon_sha256":"77f42d4b67a456cdbdc79dfb732f1196c1b46f9526c0631a62082bd7f8076271"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:31.286224Z","signature_b64":"iVM8HKMhGz8tNWGm5odzOHUle4PTK2VYLGjgNv36YxnYKmMe5lirL6dCcXnbXcFW/jjhwTOFQ9H9vS+9NIC9CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18537158073b786d18d0dba0d432225a1a3960cda71e80caedfaa75de077c140","last_reissued_at":"2026-05-18T02:27:31.285528Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:31.285528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of entire solutions to a fractional Liouville equation in $\\mathbb{R}^n$","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder","submitted_at":"2015-02-09T21:09:50Z","abstract_excerpt":"We study the existence of solutions to the problem $$ (-\\Delta)^{\\frac{n}{2}}u = Qe^{nu}\\quad\\text{in }\\mathbb{R}^n, \\quad V := \\int_{\\mathbb{R}^n}e^{nu}dx < \\infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension $n\\geq 3$ we show that to a certain extent the asymptotic behavior of $u$ and the constant $V$ can be prescribed simultaneously. Furthermore if $Q=-(n-1)!$ then $V$ can be chosen to be any positive number. 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