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The sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{\\alpha,\\beta,\\lambda,\\mu}(\\Omega) \\Big(\\int_\\Omega \\big(\\lambda \\frac{|u|^{2^*(s)}}{|x|^s}+\\mu \\frac{|v|^{2^*(s)}}{|x|^s}+2^*(s)\\kappa \\frac{|u|^\\alpha |v|^\\beta}{|x|^s}\\big)dx\\Big)^{\\frac{2}{2^*(s)}}$$  $$\\leq \\int_\\Omega \\big(|\\nabla u|^2+|\\nabla v|^2\\big)dx$$ for $(u,v)\\in {\\mathscr{D}}:=D_{0}^{1,2}(\\Omega)\\times D_{0}^{1,2}(\\Omega)$ will be explored. Further results about the sharp constant $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":"1711.10477","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-27T19:31:58Z","cross_cats_sorted":[],"title_canon_sha256":"f19974683d17daf7d9bf391a7d5d3103ace6c2352b21b42fc780553fe4322fa4","abstract_canon_sha256":"1c4bd1a0e3e0084451d1c32adf0cf9b144fbfb96af7ac1607bb7a742af42fb34"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:21.758176Z","signature_b64":"L6o7FGD858/iHzZiK93kxMrQ8dwcpgaN/uDlg+OQtFGZ/FPC1ZsVaWlmI31WhZf/Dx6JKukgKY48tMZZNKiFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e18591e0c6340170c4b99f8d5fb7aa8021a8d4ab642ad7b549c2907a41df7ca","last_reissued_at":"2026-05-18T00:29:21.757250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:21.757250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a double-variable inequality and elliptic systems involving critical Hardy-Sobolev exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2017-11-27T19:31:58Z","abstract_excerpt":"Let $\\Omega\\subset \\mathbb{R}^N$ ($N\\geq 3$) be an open domain which is not necessarily bounded. 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