Some weighted Poincar\'e inequalities

Using stable solutions of suitable PDEs, we obtain some weighted Poincaré inequalities, with applications to the Laplace operator in the Euclidean space, to the real part of the Laplace-Kohn operator in the Heisenberg group, to the sublaplacian in the Engel group, to the Franchi-Grushin- Lanconelli operators, and the p-laplacian in the Euclidean space.