Nonlocal singular problem and associated Sobolev type inequality with extremal in the Heisenberg group

We study a fractional $p$-Laplace equation involving a variable exponent singular nonlinearity in the framework of the Heisenberg group. We first establish the existence and regularity of weak solutions. In the case of a constant singular exponent, we further prove the uniqueness of solutions and characterize the extremals of a related Sobolev-type inequality. Additionally, we demonstrate a connection between the solutions of the singular problem and these extremals. To the best of our knowledge, these findings provide new insights even in the classical case $p=2$.