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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 $0\leq β<1$ and $0<γ\leq1-β$, then there holds for any $τ>0$, $$\sup_{u\in W^{1,N}(\mathbb{R}^N),\,\int_{\mathbb{R}^N}(|\nabla u|^N+τ|u|^N)dx\leq 1}\int_{\mathbb{R}^N}\frac{1}{|x|^{Nβ}}\left(e^{α_Nγ|u|^{\frac{N}{N-1}}}-\sum_{k=0}^{N-2}\frac{α_N^kγ^k|u|^{\frac{kN}{N-1}}} {k!}\right)dx<\infty,$$ where $α_N=Nω_{N-1}^{1/(N-1)}$ and $ω_{N-1}$ is the area of the unit sphere in $\mathbb{R}^N$. The above inequality is sharp in the sense that if $γ>1-β$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $β=0$ 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 $0<β<1$ and prove that for all $τ>0$, $0<β<1$ and $0<γ\leq 1-β$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.