Remarks on the Extremal Functions for the Moser-Trudinger Inequalities

2005/04/15 by Yuxiang Li, Li, Yuxiang · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP #msc:58J05

paper · pdf · doi:10.48550/arxiv.math/0504317

9 pages

arxiv created 2005/07/28 · arxiv updated 2009/12/01

Abstract

We will show in this paper that if λ is very close to 1, then I(M,λ,m)= sup_{u∈ H^1,n0(M) ,∫M|∇ u|ⁿdV=1}∫_Ω(e^{α_n |u|^(n)/(n-1)}-λ∑_k=1^m\frac|α_nu^(n)/(n-1)|^k k!)dV, can be attained, where M is a compact manifold with boundary. This result gives a counter example to the conjecture of de Figueiredo, do ó, and Ruf in their paper titled "On a inequality by N.Trudinger and J.Moser and related elliptic equations" (Comm. Pure. Appl. Math.,\bf 55:135-152, 2002).

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