  We provide new results on the existence, non-existence, localization and
multiplicity of nontrivial solutions for systems of Hammerstein integral
equations. Some of the criteria involve a comparison with the spectral radii of
some associated linear operators. We apply our results to prove the existence
of multiple nonzero radial solutions for some systems of elliptic boundary
value problems subject to nonlocal boundary conditions. Our approach is
topological and relies on the classical fixed point index. We present an
example to illustrate our theory.
