Schrodinger-Hardy system without Ambrosetti-Rambinowitz condition on Carnot groups

In this paper, we study the following Schrodinger-Hardy system {-triangle(G)u - mu psi(2)/r(xi)(2 )u = F-u(xi, u, v) in ohm, -triangle(G)v - nu psi(2)/r(xi)(2 )v = F-v(xi, u, v) in ohm, u = v = 0 on partial derivative ohm, where ohm is a smooth bounded domain on Carnot groups G, whose homogeneous dimension is Q >= 3, triangle(G )denotes the sub-Laplacian operator on G, mu and nu are real parameters, r(xi) is the natural gauge associated with fundamental solution of -triangle(G) on G, psi is the geometrical function defined as psi = |del(G (R))|, and del(G) is the horizontal gradient associated with triangle G. The difficulty is not only the nonlinearities F-u and F-v without Ambrosetti-Rabinowitz condition, but also the hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework.