Critical problems with Hardy potential on Stratified Lie Groups

Abstract

We prove existence and nonexistence results for positive solutions to the subelliptic Brezis-Nirenberg type problem with Hardy potential $$- \Delta_{\mathbb G} u -\mu \frac{\psi^2}{d^2} u =u^{2^*-1}+\lambda u\,\,\, \mbox{in}\,\,\, \Omega,\quad u=0\,\,\, \mbox{on}\,\, \partial \Omega, $$ extending to the Stratified setting well-known Euclidean results by Jannelli [J. Diff. Equ. 156, 1999]. Here, $\Delta_{\mathbb G}$ is a sub-Laplacian on an arbitrary Carnot group ${\mathbb G}$, $\Omega$ is a bounded domain of ${\mathbb G}$, $0\in \Omega$, $d$ is the $\Delta_{\mathbb G}$-gauge, $\psi:=|\Delta_{\mathbb G} d|$, where $\Delta_{\mathbb G}$ is the horizontal gradient associated to $\Delta_{\mathbb G}$, $0 \leq \mu < \bar \mu$, where $\bar \mu =\left ( \frac{Q-2}{2} \right )^2$ is the best Hardy constant on ${\mathbb G}$ and $\lambda\in {\mathbb R}$. The main difficulty in this abstract framework is the lack of knowledge of the ground state solutions to the limit problem $$- \Delta_{\mathbb G} u -\mu \dfrac{\psi^2}{d^2} u =u^{2^*-1}\,\, \mbox{on}\,\, {\mathbb G}, $$ whose explicit expression is not known, except for the case when $\mu=0$ and ${\mathbb G}$ is a group of Iwasawa-type. So, a preliminary refined analysis of qualitative properties of solutions to the above problem on the whole space is required, which has independent interest. In particular, Lorentz regularity and a priori decay estimates are obtained.