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 $\overline{\mathbb{B}}$ and satifies $\underset{x \rightarrow 0}{\lim} h_{\epsilon}(x) |x|^{- 2 \epsilon} =1$, with $h_{\epsilon} (x) >0$ on $\overline{\mathbb{B}} \setminus \{0\}$. 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_{\epsilon} (x) e^{ 4 \pi \left(1 + \epsilon \right) {u_{\epsilon}}^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_{\epsilon} (x) e^{4 \pi (1 + \epsilon) 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 $u_0 \in C^1 (\overline{ \mathbb{B}} )$. 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 \pi u^2 } dx. \end{align*}