Compactness of extremals for singular Moser–Trudinger functionals in high dimension

The main purpose of this note is to study the compactness of extremals for the singular Moser-Trudinger inequality. More precisely, let \(\Omega \subset {\mathbb {R}}^n\), \(n\ge 2\), be a bounded open smooth domain and \(0\in \Omega \), \(W^{1,n}_{0}(\Omega )\) be the standard Sobolev space. For \(\epsilon \in [0,1)\), Csato-Roy-Nguyen [J. Diff. Equ. 270:843–882, 2021] proved that the following singular Moser-Trudinger inequality

$$\begin{aligned} \sup _{u\in W_0^{1,n}(\Omega ),\,\int _{\Omega }|\nabla u|^ndx\le 1}\int _{\Omega }\frac{e^{\alpha _n(1-\epsilon )|u|^{\frac{n}{n-1}}}-1 }{|x|^{n\epsilon }}dx \end{aligned}$$

can be achieved by a nonnegative function \(u_\epsilon \in W^{1,n}_{0}(\Omega )\) with \(\int _{\Omega }|\nabla u_\epsilon |^n dx\le 1\). Here \(\alpha _{n}=n \omega _{n-1}^{1/(n-1)}\) with \(\omega _{n-1}\) being the surface area of the \((n-1)\)-dimensional unit sphere.

Relying on above result, by blow-up analysis, we consider the compactness of function family \(\{u_\epsilon \}_{0<\epsilon <1}\) and prove, up to a subsequence, \(u_\epsilon \rightarrow u_0\) in \(W^{1,n}_0(\Omega )\cap C^0({\overline{\Omega }})\cap C_{\textrm{loc}}^{1}({\overline{\Omega }}{\setminus }\{0\})\) as \(\epsilon \rightarrow 0\), where \(u_0\) is an extremal function of the following supremum

$$\begin{aligned} \sup _{u\in W_0^{1,n}(\Omega ),\,\int _{\Omega }|\nabla u|^ndx\le 1}\int _{\Omega }(e^{\alpha _n|u|^{\frac{n}{n-1}}}-1)dx. \end{aligned}$$