Multiplicity of solutions for a class of critical Schrodinger-Poisson systems on the Heisenberg group

We deal with multiplicity of solutions to the following Schrodinger-Poisson-type system in this article: {Delta(H)u -mu(1)phi(1)u = |u(2)|u + F-u(xi, u, nu) in Omega, -Delta(H)nu + mu(2) phi(2)nu = |nu|(2)nu + F-nu(xi, u, nu) in Omega, -Delta(H)phi(1) = u(2) , -Delta(H) phi(2) =nu(2) in Omega, phi(1) = phi(2) = u = nu = 0 , in Omega, where Delta(H) is the Kohn-Laplacian and Omega is a smooth bounded region on the first Heisenberg group H-1, mu(1), and mu(2) are some real parameters, and F= (x, u, v) F-u =partial derivative F/partial derivative u, F-v = partial derivative F/partial derivative u satisfying natural growth conditions. By the limit index theory and the concentration compactness principles, we prove that the aforementioned system has multiplicity of solutions for mu(1), mu(2) < |Omega|S-1/2 where S is the best Sobolev constant. The novelties of this article are the presence of critical nonlinear term, and the system is set on the Heisenberg group.