{"paper":{"title":"The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\\R}^n$ with a Hardy term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gongbao LI, Tao Yang","submitted_at":"2019-08-07T11:33:19Z","abstract_excerpt":"In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: \\begin{equation} \\label{eq0.1} (-\\Delta)^{s}u-{\\gamma} {\\frac{u}{|x|^{2s}}}= {\\frac{{|u|}^{ {2^{*}_{s}}(\\beta)-2}u}{|x|^{\\beta}}}+ \\big [ I_{\\mu}* F_{\\alpha}(\\cdot,u) \\big](x)f_{\\alpha}(x,u),\n  \\ \\ u \\in {\\dot{H}}^s(\\R^{n}) \\end{equation} where $s \\in(0,1)$, $0\\leq \\alpha,\\beta<2s<n$, $\\mu \\in (0,n)$, $\\gamma<\\gamma_{H}$, $I_{\\mu}(x)=|x|^{-\\mu}$, $F_{\\alpha}(x,u)=\\frac{ {|u(x)|}^{ {2^{\\#}_{\\mu} }(\\alpha)} }{ {|x|}^{ {\\delta_{\\mu} (\\alpha)} } }$, $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02536","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.02536/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"}