  We investigate a class of systems of partial differential equations with
nonlinear cross-diffusion and nonlocal interactions, which are of interest in
several contexts in social sciences, finance, biology, and real world
applications. Assuming a uniform "coerciveness" assumption on the diffusion
part, which allows to consider a large class of systems with degenerate
cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions
previously considered in the literature, we prove global-in-time existence of
weak solutions by means of a semi-implicit version of the
Jordan-Kinderlehrer-Otto scheme. Our approach allows to consider nonlocal
interaction terms not necessarily yielding a formal gradient flow structure.
