Ground-state solutions of a Hartree-Fock type system involving critical Sobolev exponent

In this paper, ground-state solutions to a Hartree-Fock type system with a critical growth are studied. Firstly, instead of establishing the local Palais-Smale (P.- S.) condition and estimating the mountain-pass critical level, a perturbation method is used to recover compactness and obtain the existence of ground-state solutions. To achieve this, an important step is to get the right continuity of the mountain-pass level on the coefficient in front of perturbing terms. Subsequently, depending on the internal parameters of coupled nonlinearities, whether the ground state is semi-trivial or vectorial is proved.