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

TL;DR

This paper proves the existence of extremal functions for a singular Moser-Trudinger inequality involving weights and extends previous results by filling gaps in earlier proofs, using n-harmonic transplantation techniques.

Contribution

It establishes the attainment of supremum in the weighted Moser-Trudinger inequality for all domains and addresses gaps in prior proofs for the unweighted case.

Findings

Supremum is attained on any domain for the weighted inequality.

Fills gaps in the proof for the unweighted case ($eta=0$).

Introduces n-harmonic transplantation methods for extremal analysis.

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 $\beta\in [0,n)$, $$ \sup_{u\in\mathcal{B}_1}\int_{\Omega}\frac{e^{\alpha |u|^{n/(n-1)}}}{|x|^{\beta}}\leq C \ \Leftrightarrow \ \frac{\alpha}{\alpha_n}+\frac{\beta}{n}\leq1,\qquad $$ where $\alpha_n=n\cnn$ and $\mathcal{B}_1 = \left\{ u \in W_0^{1, n}(\Omega) \ | \ \int_{\Omega} |\nabla u |^n \leq1 \right\}$. 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 $\beta=0.$