  We present a new approach to solve a Schr\"odinger Equation autonomous at
infinity, by identifying the relation between the arrangement of the spectrum
of the concerned operator and the behavior of the nonlinearity at zero and at
infinity. In order to apply variational methods, we set up a suitable linking
structure depending on the growth of the nonlinear term and making use of
information about the autonomous problem at infinity. Our method allows us to
circumvent the lack of compactness. The main novelty is that none monotonicity
assumption is required on the nonlinearity, which may be sign-changing as well
as the potential. Furthermore, depending on the nonlinearity, the limit of the
potential at infinity may be non-positive, so that zero may be an interior
point in the essential spectrum of the Schr\"odinger operator.
