Yamabe type equations with signchanging nonlinearities on the Heisenberg group, and the role of Green functions

In this paper, we investigate the existence problem for positive solutions of Yamabe type equations on the Heisenberg group Hn, driven by the Kohn-Spencer sublaplacian. The relevance of our results lies in the fact that the coefficient of the nonlinearity is allowed to change sign. A prototype case of the PDE under investigation comes from 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 to the origin. Two proofs of our main Theorem, focused on dfferent 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.