# 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…

## 47 Citations

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

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### Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

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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…

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

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- 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…

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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…

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

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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…

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is…

### Rearrangements in Carnot Groups

- MathematicsActa 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…

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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…

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- Mathematics
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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…