Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation

Abstract

The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, \textit{NoDEA Nonlinear Differential Equations Appl.}, 13 (2007), no. 5-6, 585--603] to the following weighted version: if $\Omega \subset {\mathbb{R}}^n$ is bounded, ${\omega }_{n-1}$ is the ${\mathcal{H}}^{n-1}$ measure of the unit sphere, then for $\alpha >0$ and $\beta \in [0,n)$, $$\underset{u\in {\mathcal{B}}_1}{\sup }{\int }_{\Omega }\frac{e^{\alpha |u{|}^{n/(n-1)}}}{|x{|}^{\beta }}\le C\iff \frac{\alpha }{{\alpha }_n}+\frac{\beta }{n}\le 1,$$ where $α_n=n\cnn$ and ${\mathcal{B}}_1=\left\{u\in W_0^{1,n}(\Omega )|{\int }_{\Omega }|\nabla u{|}^n\le 1\right\}$. We prove that the supremum is attained on any domain $\Omega$. The paper also fills in the gaps in the proof of [Lin K.C., Extremal functions for Moser's inequality, \textit{Trans. of. Am. Math. Soc.}, 384 (1996), 2663--2671], which deals with the case $\beta =0.$

Few minor changes are made. arXiv admin note: substantial text overlap with arXiv:1410.8638