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2017 On weak continuity of the Moser functional in Lorentz-Sobolev spaces
Robert Černý
Rocky Mountain J. Math. 47(3): 757-788 (2017). DOI: 10.1216/RMJ-2017-47-3-757

Abstract

Let $B(R)\subset \mathbb{R}^n $, $n\in \mathbb{N} $, $n\geq 2$, be an open ball. By a result from~\cite {AdT}, the Moser functional with the borderline exponent from the Moser-Trudinger inequality fails to be sequentially weakly continuous on the set of radial functions from the unit ball in $W_0^{1,n}(B(R))$, only in the exceptional case of sequences acting like a~concentrating Moser sequence.

We extend this result into the Lorentz-Sobolev space $W_0^1L^{n,q}(B(R))$, with $q\in (1,n]$, equipped with the norm $$ ||\nabla u||_{n,q}:= ||t^{1/n-1/q}|\nabla u|^*(t)||_{L^q((0,|B(R)|))}. $$ We also consider the case of a nontrivial weak limit and the corresponding Moser functional with the borderline exponent from the concentration-compactness alternative.

Citation

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Robert Černý. "On weak continuity of the Moser functional in Lorentz-Sobolev spaces." Rocky Mountain J. Math. 47 (3) 757 - 788, 2017. https://doi.org/10.1216/RMJ-2017-47-3-757

Information

Published: 2017
First available in Project Euclid: 24 June 2017

zbMATH: 1381.26020
MathSciNet: MR3682148
Digital Object Identifier: 10.1216/RMJ-2017-47-3-757

Subjects:
Primary: 26D10, 46E30, 46E35

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 3 • 2017
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