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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 Lions (Rev Mat Iberoam 1:145–201, 1985) 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}(mathbb   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 Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space (mathbb {H}^n) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polya–Szego inequality. Due to the absence of the Polya–Szego 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}): $$begin{aligned} -mathrm {div}left( left| nabla _{mathbb {H}}uright| ^{Q-2} nabla _{mathbb {H}}uright) +V(xi ) left| uright| ^{Q-2}u=frac{f(u) }{rho (xi )^{beta }} end{aligned}$$with nonlinear terms f of maximal exponential growth (exp (alpha t^{frac{Q}{Q-1}})) as (trightarrow +infty ). All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.

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