Symmetry of extremal functions in Moser-Trudinger inequalities and a Hénon type problem in dimension two
type: Item
instance of: scholarly article
title: Symmetry of extremal functions in Moser-Trudinger inequalities and a Hénon type problem in dimension two
zbMATH Open document ID: 1173.35044
author: Denis Bonheure
author: Enrico Serra
author: Massimo Tarallo
published in: Advances in Differential Equations
publication date: 2008-11-24T00:00:00Z
review text: The authors investigate the symmetry properties of extremal functions for certain maximization problems related to Moser-Trudinger inequalities. The symmetry of maximizers of a Hénon type functional in dimension two is considered, and the maximization problem \[ S_{\alpha , \gamma}=\sup_{\substack{ u\in H^1(\Omega)\\ \|u\|\leq1 }} \int_\Omega (e^{\gamma u^2}-1)|x|^{\alpha}\,dx, \] is studied, where \(\Omega\) is the unit ball of \(\mathbb R^2\), \(\alpha\) and \(\gamma\) are positive real numbers and \(\|u\|\) denotes the usual \(H^1(\Omega)\) norm. Moreover, the limit problem \[ S_{\gamma}= \sup_{\substack{ u\in H^1(\Omega)\\ \|u\|\leq1 }} \int_{\partial\Omega} (e^{\gamma u^2}-1) \,d{\sigma}, \] is investigated, which is the main ingredient to describe the behavior of maximizers as \(\alpha\to\infty\). The limit problem as \(\alpha\to 0\) and the properties of its maximizers are also discussed. The main results in this paper are the Moser-Trudinger trace inequality and the picture of the symmetry properties of solutions for different values of \(\gamma\). This interesting survey also incudes some existence and unique results, the asymptotic analysis for the maximization problem, a first symmetry breaking result, Moser-Trudinger type inequalities with weight for the limit problem, a conjecture to deserve further studies. These are the extension to dimension two of the corresponding results in dimension \(n\geq 3\).
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 5372150
zbMATH Keywords: symmetry properties
zbMATH Keywords: maximizers
zbMATH Keywords: Moser-Trudinger inequalities
zbMATH Keywords: Hénon type problem in dimension two
reviewed by: Zuoliang Xu
MaRDI profile type: Publication
Recommended article: A Trudinger-Moser type inequality and its extremal functions in dimension two
Recommended article: Extremal functions for the Trudinger-Moser inequality in 2 dimensions
Recommended article: Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities
Recommended article: Q5381419
Recommended article: Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions
Recommended article: Extremal functions for the modified Trudinger-Moser inequalities in two dimensions
Recommended article: Trudinger-Moser type inequalities with a symmetric conical metric and a symmetric potential
Recommended article: Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality
Recommended article: Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two
Recommended article: HARDY–SOBOLEV–MAZ'YA INEQUALITIES: SYMMETRY AND BREAKING SYMMETRY OF EXTREMAL FUNCTIONS
description: scientific article; zbMATH DE number 5372150