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_\epsilon(x) e^{ \alpha_N \left( 1 + \epsilon \right) |u_{\epsilon}|^{ \frac{N}{N-1} } } dx, \quad u_{\epsilon} \in W_0^{1,N} ( \mathbb{B} ) \cap \mathcal{S}, \end{align*} where $\alpha_N = \omega_N^{ \frac{1}{N-1} }$, $\omega_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_{\epsilon} \}_{\epsilon} $ and prove that it has a subsequence converging to a function in $C^1 \left(\overline{ \mathbb{B}} \right)$.
From: Qi Xia [view email]
[v1]
Wed, 24 Jun 2026 04:00:47 UTC (13 KB)
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