{"paper":{"title":"Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaomeng Li, Yunyan Yang","submitted_at":"2016-12-25T08:11:04Z","abstract_excerpt":"In a previous work (Int. Math. Res. Notices 13 (2010) 2394-2426), Adimurthi-Yang proved a singular Trudinger-Moser inequality in the entire Euclidean space $\\mathbb{R}^N$ $(N\\geq 2)$. Precisely, if $0\\leq \\beta<1$ and $0<\\gamma\\leq1-\\beta$, then there holds for any $\\tau>0$, $$\\sup_{u\\in W^{1,N}(\\mathbb{R}^N),\\,\\int_{\\mathbb{R}^N}(|\\nabla u|^N+\\tau |u|^N)dx\\leq 1}\\int_{\\mathbb{R}^N}\\frac{1}{|x|^{N\\beta}}\\left(e^{\\alpha_N\\gamma|u|^{\\frac{N}{N-1}}}-\\sum_{k=0}^{N-2}\\frac{\\alpha_N^k\\gamma^k|u|^{\\frac{kN}{N-1}}} {k!}\\right)dx<\\infty,$$ where $\\alpha_N=N\\omega_{N-1}^{1/(N-1)}$ and $\\omega_{N-1}$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08247","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"}