  We consider a semilinear elliptic problem with a nonlinear term which is the
product of a power and the Riesz potential of a power. This family of equations
includes the Choquard or nonlinear Schroedinger--Newton equation. We show that
for some values of the parameters the equation does not have nontrivial
nonnegative supersolutions in exterior domains. The same techniques yield
optimal decay rates when supersolutions exists.
