{"paper":{"title":"Minimization problem associated with an improved Hardy-Sobolev type inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Megumi Sano","submitted_at":"2019-08-11T14:36:38Z","abstract_excerpt":"We consider the existence and the non-existence of a minimizer of the following minimization problems associated with an improved Hardy-Sobolev type inequality introduced by Ioku. $$ I_a := \\inf_{u \\in W_0^{1,p}(B_R ) \\setminus \\{ 0\\} } \\frac{\\int_{B_R} |\\nabla u |^{p} \\,dx}{\\left( \\int_{B_R} |u|^{p^*(s)} V_a(x) \\,dx \\right)^{\\frac{p}{p^*(s)}}}, \\,\\,\\text{where}\\,\\, V_a (x) =\\frac{1}{|x|^s \\left( 1- a \\,\\left( \\frac{|x|}{R} \\right)^{\\frac{N-p}{p-1}} \\right)^\\beta} \\ge \\frac{1}{|x|^s}. $$ Only for radial functions, the minimization problem $I_a$ is equivalent to it associated with the classical"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03915","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03915/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}