  In this paper new criteria are established for the existence of positive
radial solutions of a semilinear elliptic system depending on the gradient.
These criteria are determined by some relationships between the upper and lower
bounds on suitable stripes of $\mathbb R^n$ of the nonlinearities of the system
and the principal characteristic values of some associated linear Hammerstein
integral operators. Moreover, using smoothing tools, the totality of the
involved cone is established.
