Multiple positive solutions for Schrodinger-Poisson system with singularity on the Heisenberg group

In this work, we study the following Schrodinger-Poisson system {-Delta(H)u + mu phi u = lambda u(-gamma), in Omega, -Delta(H)phi = u(2), in Omega, u > 0, in Omega, u = phi = 0, on partial derivative Omega, where Delta(H) is the Kohn-Laplacian on the first Heisenberg group H-1, and Omega subset of H-1 is a smooth bounded domain, mu = +/- 1, 0 < gamma < 1, and lambda > 0 are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for mu = 1 and each lambda > 0. Multiple solutions of the system are also considered for mu = -1 and lambda > 0 small enough using the critical point theory for nonsmooth functional.