Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space

2016/12/25 by LI Xiao-meng, Li, Xiaomeng, Yunyan Yang +1 · 6 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1612.08247

openalex publication_date 2016/12/25 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

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 ℝ^N (N≥ 2). Precisely, if 0≤ β<1 and 00, sup_u∈ W^1,N(ℝ^N), ∫_ℝ^N(|∇ u|^N+τ|u|^N)dx≤ 1∫_ℝ^N\frac1|x|^Nβ(e^α_Nγ|u|^(N)/(N-1)-∑_k=0^N-2\fracα_N^kγ^k|u|^(kN)/(N-1) k!)dx1-β, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β=0 has been considered by Li-Ruf (Indiana Univ. Math. J. 57 (2008) 451-480) and Ishiwata (Math. Ann. 351 (2011) 781-804). We shall investigate the singular case 00, 0

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