A fractional Moser-Trudinger type inequality in one dimension and its critical points.
type: Item
instance of: scholarly article
title: A fractional Moser-Trudinger type inequality in one dimension and its critical points.
zbMATH Open document ID: 1363.26011
author: Stefano Iula
author: Ali Maalaoui
author: Luca Martinazzi
published in: Differential and Integral Equations
publication date: 2016-04-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1504.04862
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 6562185
zbMATH Keywords: Moser-Trudinger-type inequality
zbMATH Keywords: fractional derivative
zbMATH Keywords: Sobolev space
zbMATH Keywords: fractional Laplacian
zbMATH Keywords: Bessel potential space
reviewed by: Peter Massopust
MaRDI profile type: Publication
arXiv ID: 1504.04862
Recommended article: Fractional Trudinger-Moser type inequalities in one dimension
Recommended article: Q3381309
Recommended article: Remarks on the fractional Moser-Trudinger inequality
Recommended article: Critical and subcritical fractional Trudinger-Moser-type inequalities on \(\mathbb{R}\)
Recommended article: Moser functions and fractional Moser-Trudinger type inequalities
Recommended article: On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces
Recommended article: A Moser-Trudinger type inequality on the Orlicz fractional space
Recommended article: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity
Recommended article: Singular Trudinger-Moser inequality and fractional \(p\)-Laplace equations in \(\mathbb{R}^N\)
review text: The authors investigate fractional Moser-Trudinger-type inequalities on intervals \(I\Subset \mathbb{R}\) and show that the inequality is optimal in a certain sense. Furthermore, they establish that the equation NEWLINE\[NEWLINE(-\Delta)^\frac12 u = \lambda u \exp (\tfrac12 u^2)\qquad\text{in \(I\Subset \mathbb{R}\)},NEWLINE\]NEWLINE has a positive solution in the Bessel potential space \(\widetilde{H}^{\frac12,2} (I)\) iff \(\lambda\in (0, \lambda_1(I))\), where \(\lambda_1(I)\) denotes the first eigenvalue of \((-\Delta)^\frac12\) in \(I\), extending a previous result by \textit{Adimurthi} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 3, 393-413 (1990; Zbl 0732.35028)]. Finally, a fractional Moser-Trudinger type inequality on \(\mathbb R\) is derived.
description: scientific article; zbMATH DE number 6562185