{"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. 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_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10477","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"}