Compactness of Extremals for Singular Anisotropic Trudinger-Moser functionals on bounded domain

Abstract

In this paper, we investigate the compactness of extremal functions for a critical singular anisotropic Trudinger-Moser inequality established by Lu-Shen-Xue-Zhu\cite{ref1}. We prove by means of blow-up analysis that the extremals $u_{\beta}$ converge in $W_{0}^{1,n}(\Omega)\cap C^{1}(\overline{\Omega})$ to some function $u_{0}$ which achieves the supremum \begin{equation} \sup\limits_{u\in W_{0}^{1,n}(\Omega),\Vert u\Vert_{F(\Omega)}\leq1}\int_{\Omega}^{}e^{\tau_{n}\vert u\vert^{\frac{n}{n-1}}}dx,\notag \end{equation} as $\beta\to 0$, where $\tau_{n}=n^{\frac{n}{n-1}}\kappa_{n}^{\frac{1}{n-1}}$, $\kappa_{n}$ denotes the volume of the unit Wulff ball in $\mathbb{R}^{n}$ and $\Vert u\Vert_{F(\Omega)}$ is the anisotropic norm of $u$.