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In this paper the dependence of the best constants in Sobolev andGagliardo-Nirenberg inequalities on the precise form of the Sobolev space normis investigated. The analysis is carried out on general graded Lie groups, thusincluding the cases of $\mathbb R^n$, Heisenberg, and more general stratifiedLie groups. The Sobolev norms may be defined in terms of Rockland operators,i.e. the hypoelliptic homogeneous left-invariant differential operators on thegroup. The best constants are expressed in the variational form as well as interms of the ground state solutions of the corresponding nonlinear subellipticequations. The orders of these equations can be high depending on the Sobolevspace order in the Sobolev or Gagliardo-Nirenberg inequalities, or may befractional. Applications are obtained also to equations with lower order termsgiven by different hypoelliptic operators. Already in the case of $\mathbbR^n$, the obtained results extend the classical relations by Weinstein to awide range of nonlinear elliptic equations of high orders with elliptic loworder terms and a wide range of interpolation inequalities ofGagliardo-Nirenberg type. However, the proofs are different from those ivyWeinstein because of the impossibility of using the rearrangement inequalitiesalready in the setting of the Heisenberg group. The considered class of gradedgroups is the most general class of nilpotent Lie groups where one can stillconsider hypoelliptic homogeneous invariant differential operators and thecorresponding subelliptic differential equations.

Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations