Yamabe type equations with sign-changing nonlinearities on the Heisenberg group, and the role of Green functions

In this paper, we investigate the existence problem for positive solutions of the Yamabe type equation (Y) Delta(Hn)u + q(x)u - b(x)u(sigma) = 0, sigma > 1, on the Heisenberg group H-n, where Delta(Hn) is the Kohn-Spencer sublaplacian. The relevance of our results lies in the fact that b(x) is allowed to change sign. The above PDE is tightly related to the CR Yamabe problem on the deformation of contact forms. We provide existence of a new family of solutions sharing some special asymptotic behaviour described in terms of the Koranyi distance d(x) to the origin. Two proofs of our main Theorem, focused on different aspects, will be given. In particular, the second one relies on a function-theoretic approach that emphasizes the role of Green functions; such a method is suited to deal with more general settings, notably the Yamabe equation with sign-changing nonlinearity on non-parabolic manifolds, that will be investigated in the last part of this paper.