On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

Prosenjit Roy

doi:10.3934/dcds.2019212

Article Contents

2019, Volume 39, Issue 9: 5207-5222. Doi: 10.3934/dcds.2019212

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On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

Prosenjit Roy

Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India

Received: July 31, 2018

Revised: February 28, 2019

Published: May 2019

Part of this research is supported by Inspire programme under the contract number IFA14/MA-43

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Abstract

Moser-Trudinger inequality was generalised by Calanchi-Ruf to the following version: If $ \beta \in [0,1) $ and $ w_0(x) = |\log |x||^{\beta(n-1)} $ or $ \left( \log \frac{e}{|x|}\right)^{\beta(n-1)} $ then

$ \sup\limits_{\int_B | \nabla u|^nw_0 \leq 1 , u \in W_{0,rad}^{1,n}(w_0,B)} \;\; \int_B \exp\left(\alpha |u|^{\frac{n}{(n-1)(1-\beta)}} \;\;\; \right) dx < \infty $

if and only if $ \alpha \leq \alpha_\beta = n\left[\omega_{n-1}^{\frac{1}{n-1}}(1-\beta) \right]^{\frac{1}{1-\beta}} $ where $ \omega_{n-1} $ denotes the surface measure of the unit sphere in $ \mathbb {R}^n $. The primary goal of this work is to address the issue of existence of extremal function for the above inequality. A non-existence (of extremal function) type result is also discussed, for the usual Moser-Trudinger functional.

Keywords:

Moser Trudinger embedding,
extremal function.

Mathematics Subject Classification: Primary: 35B38, 35J20, 47N20, 26D10; Secondary: 46E35.

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