TY - JOUR AU - Zhou, Chunqin AB - In this paper, we investigate a singular Moser–Trudinger inequality involving Ln\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{n}$$\end{document} norm in the entire Euclidean space. The blow-up procedures are used for the maximizing sequence. Then we obtain the existence of extremal functions for this singular geometric inequality in whole space. In general, W1,n(Rn)↪Lq(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n)$$\end{document} is a continuous embedding but not compact. But in our case we can prove that W1,n(Rn)↪Ln(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^n({\mathbb {R}}^n)$$\end{document} is a compact embedding. Combining the compact embedding W1,n(Rn)↪Lq(Rn,|x|-sdx)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n, |x|^{-s}dx)$$\end{document} for all q≥n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\ge n$$\end{document} and 0