A class of sub-elliptic equations on the heisenberg group and related interpolation inequalities

Abstract

We firstly prove the existence of least energy solutions to a class of sub-elliptic equations on the Heisenberg group. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo-Nirenberg inequality on the Heisenberg group. Finally we point out some extensions to the quasilinear sub-elliptic case.