In this paper, we investigate a singular Moser–Trudinger inequality involving \(L^{n}\) 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, \(W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n)\) is a continuous embedding but not compact. But in our case we can prove that \(W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^n({\mathbb {R}}^n)\) is a compact embedding. Combining the compact embedding \(W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n, |x|^{-s}dx)\) for all \(q\ge n\) and \(0<s<n\) in [18], we establish the theorems for any \(0\le \alpha <1\).

在本文中，我们研究了在整个欧几里得空间中涉及\(L^{n}\)范数的奇异 Moser-Trudinger 不等式。爆破程序用于最大化序列。然后我们得到这个奇异几何不等式在整个空间中的极值函数的存在性。一般来说，\(W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n)\)是连续嵌入但不紧凑。但在我们的例子中，我们可以证明\(W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^n({\mathbb {R}}^n)\)是一个紧凑的嵌入. 结合紧凑嵌入\(W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n, |x|^{-s}dx) \)对于所有\(q\ge n\)和\(0<s<n\)在 [18] 中，我们建立了任何\(0\le \alpha <1\)的定理。