Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Concentration-compactness principle for Trudinger-Moser inequalities on
  Heisenberg Groups and existence of ground state solutions   Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional
Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We
extend the well-known concentration-compactness principle on finite domains in
the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group
$\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding
concentration-compactness principle for the Sobolev space $HW^{1,Q}\left(
\mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$.
  Our results improve the sharp Trudinger-Moser inequality on domains of finite
measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the
corresponding one on the whole space $\mathbb{H}^n$ by Lam and the second
author [21]. All the proofs of the concentration-compactness principles in the
literature even in the Euclidean spaces use the rearrangement argument and the
Poly\'a-Szeg\"{o} inequality. Due to the absence of the Poly\'a-Szeg\"{o}
inequality on the Heisenberg group, we will develop a different argument. Our
approach is surprisingly simple and general and can be easily applied to other
settings where symmetrization argument does not work. As an application of the
concentration-compactness principle, we establish the existence of ground state
solutions for a class of $Q$- Laplacian subelliptic equations on
$\mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth
$\exp\left( \alpha t^{\frac{Q}{Q-1}}\right) $ as $t\rightarrow+\infty$.

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The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph