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}
}
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… 

Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations

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

Sharp Singular Trudinger–Moser Inequalities Under Different Norms

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

Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

  • 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.

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

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

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

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}

Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

  • 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

Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials

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}

Rearrangements in Carnot Groups

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

Sharpened Trudinger–Moser Inequalities on the Euclidean Space and Heisenberg Group

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

Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations

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

A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^n$

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

Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces

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

A sharp Trudinger–Moser inequality on any bounded and convex planar domain

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

Remarks on the Extremal Functions for the Moser–Trudinger Inequality

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}

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

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

Extremal functions for the trudinger-moser inequality in 2 dimensions

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

Rearrangements in Carnot Groups

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

Sharp Constants and Optimizers for a Class of Caffarelli–Kohn–Nirenberg Inequalities

Abstract In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

AbstractWe derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the
...