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

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 $\mathbb{R}^N$ $(N\geq 2)$. Precisely, if $0\leq \beta<1$ and $0<\gamma\leq1-\beta$, then there holds for any $\tau>0$, $$\sup_{u\in W^{1,N}(\mathbb{R}^N),\,\int_{\mathbb{R}^N}(|\nabla u|^N+\tau |u|^N)dx\leq 1}\int_{\mathbb{R}^N}\frac{1}{|x|^{N\beta}}\left(e^{\alpha_N\gamma|u|^{\frac{N}{N-1}}}-\sum_{k=0}^{N-2}\frac{\alpha_N^k\gamma^k|u|^{\frac{kN}{N-1}}} {k!}\right)dx<\infty,$$ 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 $0<\gamma\leq 1-\beta$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.