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

Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups