On coupled systems of nonlinear Schrödinger and Choquard equations with distinct exponents

In this paper, we are interested in the existence of a positive solution of the two coupled system of nonlinear Schrödinger and Choquard equations. Our equations admit the case that the nonlinearity exponents of two components are different. This includes, in particular, the upper and lower critical cases. We first investigate the existence of ground states and provide a criterion for two-component positiveness of ground states. In case where the ground state is semi-trivial, we construct a higher energy positive solution of the mountain-pass type on the Nehari manifold.