The compactness of Moser-Trudinger functionals with conical metric in the unit ball of $\mathbb{R}^N$

Abstract

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$, $W_0^{1,N} \left( \mathbb{B} \right)$ is a standard Sobolev space. Zhang proved the extremal function of the Moser-Trudinger inequality as follows, \begin{align*} \int_{ \mathbb{B}} h_ε(x) e^{ α_N \left( 1 + ε\right) |u_ε|^{ \frac{N}{N-1} } } dx, \quad u_ε \in W_0^{1,N} ( \mathbb{B} ) \cap \mathcal{S}, \end{align*} where $α_N = ω_N^{ \frac{1}{N-1} }$, $ω_N $ is the area of the unit sphere in $\mathbb{R}^N$(see \citep{26}) . In this paper, we consider the compactness of the sequence $\{ u_ε \}_ε $ and prove that it has a subsequence converging to a function in $C^1 \left(\overline{ \mathbb{B}} \right)$.