A sharp Trudinger-Moser type inequality involving Ln norm in the entire space Rn

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Lu, Guozhen¹  Zhu, Maochun² 

[1] Univ Connecticut, Dept Math, Storrs, CT 06269 USA

[2] Jiangsu Univ, Fac Sci, Zhenjiang 212013, Jiangsu, Peoples R China

关键词: Trudinger-Moser inequality;    Blow up analysis;    Extremal functions;    Unbounded domain;   

DOI  :  10.1016/j.jde.2019.03.037

来源: Elsevier

【 摘 要 】

Let W-1,W-n (R-n) be the standard Sobolev space and parallel to.parallel to(n) be the L-n norm on R-n. We establish a sharp form of the following Trudinger-Moser inequality involving the L-n norm sup (parallel to u parallel to w1,n(Rn)=1) integral(Rn)Phi(alpha(n)vertical bar u vertical bar(n/n-1) (1 + alpha parallel to u parallel to(n)(n))(1/n-1)) dx < +infinity for any 0 <= alpha < 1, where Phi(t) = e(t) - Sigma(n-2)(j=0) t(j)/j!, alpha(n) = n omega(1/n-1)(n-1) and omega(n - 1) dimensional surface measure of the unit ball in R-n. We also show that the above supremum is infinity for all alpha >= 1. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when alpha > 0 is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our results sharpen the recent work [19] in which they show that the above inequality holds in a weaker form when Phi(t) is replaced by a strictly smaller Phi* (t) = e(t) - Sigma(n-1)(j=0) t(j)/j! (note that Phi (t) = Phi* (t) + t(n-1)/(n- 1)!). (C) 2019 Published by Elsevier Inc.

【 授权许可】

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