  We prove the existence of large-data global-in-time weak solutions to a
general class of coupled bead-spring chain models with finitely extensible
nonlinear elastic (FENE) type spring potentials for nonhomogeneous
incompressible dilute polymeric fluids in a bounded domain in $\mathbb{R}^d$,
$d=2$ or $3$. The class of models under consideration involves the
Navier--Stokes system with variable density, where the viscosity coefficient
depends on both the density and the polymer number density, coupled to a
Fokker--Planck equation with a density-dependent drag coefficient. The proof is
based on combining a truncation of the probability density function with a
two-stage Galerkin approximation and weak compactness and compensated
compactness techniques to pass to the limits in the sequence of Galerkin
approximations and in the truncation level.
