{"paper":{"title":"Extremal functions for the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\\mathbb R^N)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Van Hoang Nguyen","submitted_at":"2017-02-26T01:53:11Z","abstract_excerpt":"We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\\mathbb R^N)$ \\[ MT(N,\\beta, \\alpha) =\\sup_{u\\in W^{1,N}(\\mathbb R^N), \\|\\nabla u\\|_N^N + \\|u\\|_N^N\\leq 1} \\int_{\\mathbb R^N} \\Phi_N(\\beta(1+\\alpha \\|u\\|_N^N)^{\\frac1{N-1}} |u|^{\\frac N{N-1}}) dx, \\] where $\\Phi_N(t) =e^{t} -\\sum_{k=0}^{N-2} \\frac{t^k}{k!}$, $0\\leq \\alpha < 1$ both in the subcritical case $\\beta < \\beta_N$ and critical case $\\beta =\\beta_N$ with $\\beta_N = N \\omega_{N-1}^{\\frac1{N-1}}$ and $\\omega_{N-1}$ denotes the su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.07970","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":""},"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"}