This paper is focused on the solvability of a family of nonlinear elliptic systems defined in RN . Such equations contain Hardy potentials and Hardy–Sobolev criticalities coupled by a possible critical Hardy–Sobolev term. That problem arises as a generalization of Gross–Pitaevskii and Bose–Einstein type systems. By means of variational techniques, we shall find ground and bound states in terms of the coupling parameter ν and the order of the different parameters and exponents. In particular, for a wide range of parameters we find solutions as minimizers or Mountain–Pass critical points of the energy functional on the underlying Nehari manifold.
