We study the semilinear Schr\"odinger type equation

 -\Delta u +V(x)u=f(u),  \ \ \ x\in\{\bf R}^N,

under the assumption that $f$ is asymptotically linear, that is
$f(u)/u \to a<\infty$ as $|u| \to\ infty$. We will also consider the
case $f'(0)\ne 0$. The main focus of this talk is the extension of
known results for bounded domains to the whole space
$\{\bf R}^N$. In particular, the existence of signed and
multiple sign-changing (nodal) solutions. Our approach throughout
will be variational. We will highlight some of the difficulties
specific to this class of nonlinearities, and illustrate some
techniques for dealing with the lack of compactness.