  The purpose of this paper is twofold: first we study an eigenvalue problem
for the fractional $p$-sub-Laplacian over the fractional Folland-Stein-Sobolev
spaces on stratified Lie groups. We apply variational methods to investigate
the eigenvalue problems. We conclude the positivity of the first eigenfunction
via the strong minimum principle for the fractional $p$-sub-Laplacian.
Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly,
utilising established properties, we prove the existence of at least two weak
solutions via the Nehari manifold technique to a class of subelliptic singular
problems associated with the fractional $p$-sub-Laplacian on stratified Lie
groups. We also investigate the boundedness of positive weak solutions to the
considered problem via the Moser iteration technique. The results obtained here
are also new even for the case $p=2$ with $\mathbb{G}$ being the Heisenberg
group.
