On sharp singular Moser-Trudinger type inequalities and applications to zero mass $(p,Q)$-Laplace subelliptic equations in the Heisenberg group

This article establishes a sharp singular Moser-Trudinger type inequality in a new function space and explores a concentration-compactness principle in the Heisenberg group, inspired by P.\textcolor{blue}{-}L. Lions. We analyze the equivalence of sharp critical and subcritical singular Moser-Trudinger inequalities, focusing on their asymptotic behavior and connections between their suprema. Additionally, we apply the mountain pass theorem to demonstrate the existence of positive ground state solutions for zero mass $(p,Q)$-Laplace subelliptic equations with singular exponential nonlinearity for the following equation: $$-{\mathrm{\Delta }}_{H,p}u-{\mathrm{\Delta }}_{H,Q}u=\frac{f(\xi ,u)}{r(\xi {)}^{\vartheta }}\text{in}{\mathbb{H}}^n,$$ with $1<p<Q$, $\vartheta \in (0,Q)$, $Q=2n+2$, the function $r(\cdot )$ is called the Korányi norm in ${\mathbb{H}}^n$ and the nonlinearity $f:{\mathbb{H}}^n\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which behaves like $\exp (\alpha \mathrm{\mid }s{\mathrm{\mid }}^{\frac{Q}{Q-1}})$ as $\mathrm{\mid }s\mathrm{\mid }\to +\mathrm{\infty }$ for some $\alpha >0$.