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

Let $mathbb{H}^{n}=mathbb{C}^{n}timesmathbb{R}$ 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 Polya-Szeg{o} inequality. Due to the absence of the Polya-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 $expleft( alpha t^{frac{Q}{Q-1}}right) $ as $trightarrow+infty$.