  The aim of this paper is investigating the existence of solutions of some
semilinear elliptic problems on open bounded domains when the nonlinearity is
subcritical and asymptotically linear at infinity and there is a perturbation
term which is just continuous. Also in the case when the problem has not a
variational structure, suitable procedures and estimates allow us to prove that
the number of distinct crtitical levels of the functional associated to the
unperturbed problem is "stable" under small perturbations, obtaining
multiplicity results if the nonlinearity is odd, both in the non--resonant and
in the resonant case.
