## Rocky Mountain Journal of Mathematics

### On weak continuity of the Moser functional in Lorentz-Sobolev spaces

Robert Černý

#### 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.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 3 (2017), 757-788.

Dates
First available in Project Euclid: 24 June 2017

https://projecteuclid.org/euclid.rmjm/1498269810

Digital Object Identifier
doi:10.1216/RMJ-2017-47-3-757

Mathematical Reviews number (MathSciNet)
MR3682148

Zentralblatt MATH identifier
1381.26020

#### Citation

Černý, Robert. On weak continuity of the Moser functional in Lorentz-Sobolev spaces. Rocky Mountain J. Math. 47 (2017), no. 3, 757--788. doi:10.1216/RMJ-2017-47-3-757. https://projecteuclid.org/euclid.rmjm/1498269810

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