A sharp inequality of Trudinger-Moser type and extremal functions in H1,n (Rn)

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

do O, Joao Marcos¹  de Souza, Manasses¹ 

[1] Univ Fed Paraiba, Dept Math, BR-58051900 Joao Pessoa, Paraiba, Brazil

关键词: Limiting Sobolev inequalities;    Trudinger-Moser inequality;    Sharp constants;    Extremal function;    Blow-up analysis;   

DOI  :  10.1016/j.jde.2015.01.026

来源: Elsevier

【 摘 要 】

We prove a sharp form of the Trudinger-Moser inequality for the Sobolev space H-1,H-n (R-n). The sharpness comes from the introduction of an extra factor parallel to u parallel to(n)(n) in the classical Trudinger-Moser inequality. Let l(alpha):= sup({u is an element of H1,n(Rn):parallel to u parallel to 1,n = 1}) integral(Rn) (Phi o nu alpha(u)dx,) where Phi (t) := e(t) -Sigma(n-1)(i=0) t(i)/i(i) and nu(alpha) (u):= beta(n)(1 + alpha parallel to u parallel to(n)(n))(1/n-1))vertical bar u vertical bar(n/(n-1)). The main results read: (1) for 0 <= alpha < 1 we have l (alpha) < infinity, (2) for alpha > 1, l(alpha) = infinity and (3) moreover, we prove that for 0 <= alpha < 1, an extremal function for l(alpha) exists. (c) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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