  In this paper we show how quantum corrections, although perturbatively small,
may play an important role in the analysis of the existence of some classical
models. This, in fact, appears to be the case of static, uniform--density
models of the interior metric of cosmic strings and neutron stars. We consider
the fourth order semiclassical equations and first look for perturbative
solutions in the coupling constants $\alpha$ and $\beta$ of the quadratic
curvature terms in the effective gravitational Lagrangian. We find that there
is not a consistent solution; neither for strings nor for spherical stars. We
then look for non--perturbative solutions and find an explicit approximate
metric for the case of straight cosmic strings. We finally analyse the
contribution of the non--local terms to the renormalized energy--momentum
tensor and the possibility of this terms to allow for a perturbative solution.
We explicitly build up a particular renormalized energy--momentum tensor to
fulfill that end. These state--dependent corrections are found by simple
considerations of symmetry, conservation law and trace anomaly, and are chosen
to compensate for the local terms. However, they are not only ad hoc, but have
to depend on $\alpha$ and $\beta$, what is not expected to first perturbative
order. We then conclude that non--perturbative solutions are valuable for
describing certain physical situations.
