Ground states for scalar field equations with anisotropic nonlocal nonlinearities

We consider a class of scalar field equations with anisotropic non-local nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.