Existence results for critical problems involving p-sub-Laplacians on Carnot groups

We provide existence results for the quasilinear subelliptic Dirichlet problem $$-\Delta_{p, \mathbb{G}}u = |u|^{p^*-2}u + g(\xi, u) \quad \mbox{in}\, \Omega,\quad u\in S_0^{1,p}(\Omega),$$ where $\Delta_{p,\mathbb{G}}$ is the $p$-sub-Laplacian on a Carnot group $\mathbb{G}$, $p^*= pQ/(Q-p)$ is the critical Sobolev exponent in this context, $\Omega$ is a bounded domain of $\mathbb{G}$ and $g(\xi, u)$ is a subcritical perturbation. By means of standard variational methods adapted to the stratified context, we prove the existence of solutions both in the mountain pass and in the linking case. A crucial ingredient in this abstract framework is the knowledge of the exact rate of decay of the $p$-Sobolev extremals on Carnot groups.