Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group

In this paper we first prove existence of nontrivial nonnegative solutions of a Schrödinger–Hardy system in the Heisenberg group, driven by two possibly different Laplacian operators. The main originality of the paper is to work in the Heisenberg group. In fact several new theorems have to be proved in order to overcome the difficulties arising in the new framework, also due to the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Finally, we discuss and prove existence even for systems in the Heisenberg group, including critical nonlinear terms.