  In this paper, we establish the sharp fractional subelliptic Sobolev
inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The
best constants are given in terms of a ground state solution of a fractional
subelliptic equation involving the fractional $p$-sublaplacian ($1<p<\infty$)
on stratified Lie groups. We also prove the existence of ground state (least
energy) solutions to nonlinear subelliptic fractional Schr\"odinger equation on
stratified Lie groups. Different from the proofs of analogous results in the
setting of classical Sobolev spaces on Euclidean spaces given by Weinstein
(Comm. Math. Phys. 87(4):576-676 (1982/1983)) using the rearrangement
inequality which is not available in stratified Lie groups, we apply a
subelliptic version of vanishing lemma due to Lions extended in the setting of
stratified Lie groups combining it with the compact embedding theorem for
subelliptic fractional Sobolev spaces obtained in our previous paper (Math.
Ann. (2023)). We also present subelliptic fractional logarithmic Sobolev
inequalities with explicit constants on stratified Lie groups. The main results
are new for $p=2$ even in the context of the Heisenberg group.
