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

created by csato on 12 Nov 2021

[BibTeX]

preprint

Inserted: 12 nov 2021

Year: 2018

ArXiv: 1801.03932 PDF

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 $\beta\in [0,n)$, $$\sup{u\in\mathcal{B}1}\int{\Omega}\frac{e{\alpha u {n(n-1)}}}{ x {\beta}}\leq C \ \Leftrightarrow \ \frac{\alpha}{\alphan}+\frac{\beta}{n}\leq1,\qquad$$ 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 $\beta=0.$

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