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 $${\int }_{\mathbb{B}}\mathrm{\mid }x{\mathrm{\mid }}^{N\epsilon }\mathrm{\Phi }\left({\alpha }_N(1+\epsilon )\mathrm{\mid }u{\mathrm{\mid }}^{\frac{N}{N-1}}\right)dx$$ over all such functions is uniformly bounded. Furthermore, we also prove the existence of extremals. Finally, we consider the compactness of the sequence of extremals of the inequalities and the limit of this sequence is the extremal of $${\int }_{\mathbb{B}}\mathrm{\Phi }\left({\alpha }_N\mathrm{\mid }u{\mathrm{\mid }}^{\frac{N}{N-1}}\right)dx$$ in $C^1\left(\mathbb{B}\right)$, where ${\alpha }_N=N{\omega }_{N-1}^{\frac{1}{N-1}}$, $\mathrm{\Phi }\left(t\right):=e^t-{\sum }_{j=0}^{N-2}\frac{t^j}{j!}$ and ${\omega }_{N-1}$ is the surface of the unit ball in ${\mathbb{R}}^N$.