  We consider a class of scalar field equations with anisotropic nonlocal
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.
