The Schrodinger-Poisson type system involving a critical nonlinearity on the first Heisenberg group

This paper is concerned with the following Schrodinger-Poisson type system: {-Delta Hu+mu phi u=lambda divide u divide q-2u+ divide u divide 2u, in omega,-Delta H phi=u2, in omega,phi=u=0, on partial differential omega, where Delta(h) is the Kohn-Laplacian on the first Heisenberg group (1) and omega subset of (1) is a smooth bounded domain, 1 < q < 2, mu is an element of Double-struck capital R and lambda > 0 some real parameters. By the Green's representation formula, the concentration compactness and the critical point theory, we prove that the above system has at least two positive solutions for mu < S x meas(omega)(-1/2) and 1/2 small enough, where S s the best Sobolev constant. Moreover, we show also that there is a positive ground state solution for the above system. Our result is new even in the Euclidean case.