Semilinear PDEs in the Heisenberg group: The role of the right invariant vector fields

Some geometric features of the stable solutions of semilinear PDEs in the Heisenberg group are dealt with.A nonexistence result is given. The analysis is carried out by investigating the role of the right invariant vector fields: the monotonicity along these vector fields implies the stability of the solution and it naturally gives rise to geometrically interesting quantities.