The compactness of Moser-Trudinger type inequalities in the unit ball

In this paper, we employ the concentration-compactness principle to show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of $$ \int_{\mathbb{B}} |x|^{N ε} Φ\left( α_N \left( 1+ ε\right) |u|^{ \frac{N}{N-1} } \right) dx $$ over all such functions is uniformly