# Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

@article{Li2017ConcentrationcompactnessPF,
title={Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions},
author={Jungang Li and Guo-zhen Lu and Maochun Zhu},
journal={Calculus of Variations and Partial Differential Equations},
year={2017},
volume={57},
pages={1-26}
}
• Published 3 March 2017
• Mathematics
• Calculus of Variations and Partial Differential Equations
Let $$\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}$$Hn=Cn×R be the n-dimensional Heisenberg group, $$Q=2n+2$$Q=2n+2 be the homogeneous dimension of $$\mathbb {H}^{n}$$Hn. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group $$\mathbb {H}^{n}$$Hn. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $${ HW}^{1… • Mathematics • 2017 Abstract In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a • Mathematics Advanced Nonlinear Studies • 2019 Abstract The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝN{\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces • Xumin Wang • Mathematics, Computer Science Communications on Pure & Applied Analysis • 2019 The singular Hardy-Trudinger-Moser inequality is presented and the existence of extremal functions in a suitable function space is established by the method of blow-up analysis. • Mathematics Advanced Nonlinear Studies • 2018 Abstract In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ4{\mathbb{R}^{4}}. We also give a • Mathematics Science China Mathematics • 2021 The classical critical Trudinger-Moser inequality in ℝ2 under the constraint ∫ℝ2(|∇u|2+|u|2)dx⩽1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} • Xu Wang • Mathematics Acta Mathematica Sinica, English Series • 2020 In this paper, we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in ℝn, where n ≥ 2. More precisely, we show that for any given α > 0 and 0 < t < n, then the • Mathematics Calculus of Variations and Partial Differential Equations • 2020 In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in H2R4\documentclass[12pt]{minimal} \usepackage{amsmath} • Mathematics Acta Mathematica Sinica, English Series • 2019 In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br, or • Materials Science The Journal of Geometric Analysis • 2021 Let Hn=Cn×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} ## References SHOWING 1-10 OF 65 REFERENCES • Mathematics • 2017 Abstract In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a • Mathematics • 2006 The Trudinger-Moser inequality states that for functions u \in H_0^{1,n}(\Omega) (\Omega \subset \mathbb R^n a bounded domain) with \int_\Omega |\nabla u|^ndx \le 1 one has \int_\Omega • Mathematics • 2016 Abstract In this paper, we first establish a singular (0<β<n{(0<\beta<n}) Trudinger–Moser inequality on any bounded domain in ℝn{\mathbb{R}^{n}} with Lorentz–Sobolev norms (Theorem 1.1). Next, we • Mathematics • 2015 Wang and Ye conjectured in (Adv Math 230:294–320, 2012): Let$$\Omega $$Ωbe a regular, bounded and convex domain in$$\mathbb {R}^{2}$$R2. There exists a finite constant$$C({\Omega })>0$$C(Ω)>0such AbstractWe will show in this paper that if λ is very close to 1, then$$ I(M,\lambda ,m) = {\mathop {\sup }\limits_{u \in H^{{1,n}}_{0} (M),\smallint _{M} {\left| {\nabla u} \right|}^{n} dV = 1}
• Mathematics
AbstractWe prove that theTrudinger-Moser constant $$\sup \left\{ {\int_\Omega {\exp (4\pi u^2 )dx:u \in H_0^{1,2} (\Omega )\int_\Omega {\left| {\nabla u} \right|^2 dx \leqslant 1} } } \right\}$$ is