{"paper":{"title":"Borderline variational problems involving fractional Laplacians and critical singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nassif Ghoussoub, Shaya Shakerian","submitted_at":"2015-03-27T19:32:42Z","abstract_excerpt":"We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \\begin{equation*} \\mu_{\\gamma,s}(\\R^n):= \\inf\\limits_{u \\in H^{\\frac{\\alpha}{2}} (\\R^n)\\setminus \\{0\\}} \\frac{ \\int_{\\R^n} |({-}{ \\Delta})^{\\frac{\\alpha}{4}}u|^2 dx - \\gamma \\int_{\\R^n} \\frac{|u|^2}{|x|^{\\alpha}}dx }{(\\int_{\\R^n} \\frac{|u|^{2_{\\alpha}^*(s)}}{|x|^{s}}dx)^\\frac{2}{2_{\\alpha}^*(s)}}, \\end{equation*} where $0\\leq s<\\alpha<2$, $n>\\alpha$, ${2_{\\alpha}^*(s)}:=\\frac{2(n-s)}{n-{\\alpha}},$ and $\\gamma \\in \\mathbb{R}$. This allows us to establish the existence of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08193","kind":"arxiv","version":2},"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"}