The compactness of Moser-Trudinger functionals with conical metric in the unit ball

Let $\mathbb{B}$ be the unit ball in ${\mathbb{R}}^2$, $W_0^{1,2}\left(\mathbb{B}\right)$ is a standard Sobolev space. Suppose a function $h_{\epsilon }(x)$ is radially symmetric, nonnegative, continuous on $\bar{\mathbb{B}}$ and satifies $h_{\epsilon }(x)\mathrm{\mid }x{\mathrm{\mid }}^{-2\epsilon }=1$, with $h_{\epsilon }(x)>0$ on $\bar{\mathbb{B}}\setminus \{0\}$. In \citep{26}, Zhang proved that the supremum in the following inequality can be attained by some function $u_{\epsilon }$, i.e. , \begin{align} \int_{ $\mathbb{B}$ } h_ε (x) e^{ 4 π\left(1 + ε\right) {u_ε}^2 } dx = \underset{u \in W_0^{1,2} \left( $\mathbb{B}$ \right) \cap $\mathcal{S}$ \setminus {0} , ~ \int_{ $\mathbb{B}$ } |\nabla u|^2 dx \leq 1}{\sup} \int_{$\mathbb{B}$} h_ε (x) e^{4 π(1 + ε) u^2 } dx, \label{eq: 0.1} \end{align} where $4\pi$ is the best constant in the classical Moser-Trudinger inequality, and $\mathcal{S}$ is the set of radially symmetric functions. In this paper, we consider the compactness of the sequence $\{u_{\epsilon }{\}}_{\epsilon }$ and prove that the limit of this sequence is a function $u_0\in C^1(\bar{\mathbb{B}})$. Moreover, the $u_0$ is an extremal function of the supremum \begin{align*} \underset{u \in W_0^{1,2} \left( $\mathbb{B}$ \right) \cap $\mathcal{S}$ \setminus {0} , ~ \int_{ $\mathbb{B}$ } |\nabla u|^2 dx \leq 1}{\sup} \int_{$\mathbb{B}$} e^{4 πu^2 } dx. \end{align*}