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}(Ω),\Vert u\Vert_{F(Ω)}\leq1}\int_Ω^{}e^{τ_{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 $u$.