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

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 ${\mathbb{R}}^N$ $(N\ge 2)$. Precisely, if $0\le \beta <1$ and $0<\gamma \le 1-\beta$, then there holds for any $\tau >0$, $$\underset{u\in W^{1,N}({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}(|\nabla u{|}^N+\tau |u{|}^N)dx\le 1}{\sup }{\int }_{{\mathbb{R}}^N}\frac{1}{|x{|}^{N\beta }}\left(e^{{\alpha }_N\gamma |u{|}^{\frac{N}{N-1}}}-\sum _{k=0}^{N-2}\frac{{\alpha }_N^k{\gamma }^k|u{|}^{\frac{kN}{N-1}}}{k!}\right)dx<\infty ,$$ where ${\alpha }_N=N{\omega }_{N-1}^{1/(N-1)}$ and ${\omega }_{N-1}$ is the area of the unit sphere in ${\mathbb{R}}^N$. The above inequality is sharp in the sense that if $\gamma >1-\beta$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $\beta =0$ 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 $0<\beta <1$ and prove that for all $\tau >0$, $0<\beta <1$ and $0<\gamma \le 1-\beta$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.

35 pages

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