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2020 When does a perturbed Moser–Trudinger inequality admit an extremal?
Pierre-Damien Thizy
Anal. PDE 13(5): 1371-1415 (2020). DOI: 10.2140/apde.2020.13.1371

Abstract

We are interested in several questions raised mainly by Mancini and Martinazzi (2017) (see also work of McLeod and Peletier (1989) and Pruss (1996)). We consider the perturbed Moser–Trudinger inequality Iαg(Ω) at the critical level α=4π, where g, satisfying g(t)0 as t+, can be seen as a perturbation with respect to the original case g0. Under some additional assumptions, ensuring basically that g does not oscillate too fast as t+, we identify a new condition on g for this inequality to have an extremal. This condition covers the case g0 studied by Carleson and Chang (1986), Struwe (1988), and Flucher (1992). We prove also that this condition is sharp in the sense that, if it is not satisfied, I4πg(Ω) may have no extremal.

Citation

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Pierre-Damien Thizy. "When does a perturbed Moser–Trudinger inequality admit an extremal?." Anal. PDE 13 (5) 1371 - 1415, 2020. https://doi.org/10.2140/apde.2020.13.1371

Information

Received: 6 February 2018; Revised: 13 March 2019; Accepted: 12 May 2019; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07271833
MathSciNet: MR4149065
Digital Object Identifier: 10.2140/apde.2020.13.1371

Subjects:
Primary: 35B33 , 35B44 , 35J15 , 35J61

Keywords: blow-up analysis , elliptic equations , extremal function , Moser–Trudinger inequality

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 5 • 2020
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