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

Abstract

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 $H{W}^{1,Q}\left({\mathbb{H}}^n\right)$ 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 Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö 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 $t\to +\infty$.

28 pages

References in corpus (1)

• Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities