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In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type andBrezis-Gallouet-Wainger inequalities associated with the positive Rocklandoperators on graded Lie groups, which includes the cases of $\mathbb R^n$,Heisenberg, and general stratified Lie groups. As an application, using thecritical Gagliardo-Nirenberg inequality, the existence of least energysolutions of nonlinear Schr\"{o}dinger type equations is obtained. We alsoexpress the best constant in the critical Gagliardo-Nirenberg and Trudingerinequalities in the variational form as well as in terms of the ground statesolutions of the corresponding nonlinear subelliptic equations. The obtainedresults are already new in the setting of general stratified Lie groups(homogeneous Carnot groups). Among new technical methods, we also extendFolland's analysis of H\"older spaces from stratified Lie groups to generalhomogeneous Lie groups.

Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states