  In this paper the dependence of the best constants in Sobolev and
Gagliardo-Nirenberg inequalities on the precise form of the Sobolev space norm
is investigated. The analysis is carried out on general graded Lie groups, thus
including the cases of $\mathbb R^n$, Heisenberg, and more general stratified
Lie groups. The Sobolev norms may be defined in terms of Rockland operators,
i.e. the hypoelliptic homogeneous left-invariant differential operators on the
group. The best constants are expressed in the variational form as well as in
terms of the ground state solutions of the corresponding nonlinear subelliptic
equations. The orders of these equations can be high depending on the Sobolev
space order in the Sobolev or Gagliardo-Nirenberg inequalities, or may be
fractional. Applications are obtained also to equations with lower order terms
given by different hypoelliptic operators. Already in the case of $\mathbb
R^n$, the obtained results extend the classical relations by Weinstein to a
wide range of nonlinear elliptic equations of high orders with elliptic low
order terms and a wide range of interpolation inequalities of
Gagliardo-Nirenberg type. However, the proofs are different from those ivy
Weinstein because of the impossibility of using the rearrangement inequalities
already in the setting of the Heisenberg group. The considered class of graded
groups is the most general class of nilpotent Lie groups where one can still
consider hypoelliptic homogeneous invariant differential operators and the
corresponding subelliptic differential equations.
