  We prove new results on the existence, non-existence, localization and
multiplicity of nontrivial radial solutions of a system of elliptic boundary
value problems on exterior domains subject to nonlocal, nonlinear, functional
boundary conditions. Our approach relies on fixed point index theory. As a
by-product of our theory we provide an answer to an open question posed by do
{\'O}, Lorca, S\'{a}nchez and Ubilla. We include some examples with explicit
nonlinearities in order to illustrate our theory.
