Critical nonlocal Schrodinger-Poisson system on the Heisenberg group

In this paper, we are concerned with the following a new critical nonlocal Schrodinger-Poisson system on the Heisenberg group: {-(a - b integral Omega vertical bar del(H)u vertical bar(2)d xi) Delta(H)u + mu phi u = lambda vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(2) u, in Omega, -Delta(H)phi = u(2), 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, a, b > 0, 1 < q < 2 or 2 < q < 4, lambda > 0 and mu epsilon R are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several diffculties arising in the framework of Heisenberg groups, also due to the presence of the non-local coeffcient (a-b integral Omega vertical bar del(H)u vertical bar(2) dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.