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In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [ 25 ], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality inℝ  n{\mathbb{R}^{n}}   . Then we prove a compact embedding theorem, which states thatW   1  ,  n    ⁢   (   ℝ  n   ){W^{1,n}(\mathbb{R}^{n})}    is compactly embedded intoL  p   ⁢   (   ℝ  n   ,|  x  |    -  β⁢  d  ⁢  x   ){L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)}    forp  ≥  n{p\geq n}    and0  &lt;  β  &lt;  n{0&lt;\beta&lt;n}   . As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the following  n -Laplace equation with critical nonlinearity:

Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝ n and n -Laplace Equations