  This paper concerns the existence of a nontrivial solution for the following
problem
  \begin{equation}
  \left\{\begin{aligned}
  -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e.
in}\;\;\mathbb{R}^{N},\nonumber
  u \in H^{1}(\mathbb{R}^{N}),
  \end{aligned}
  \right.\leqno{(P)}
  \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a discontinuous
function and asymptotically linear at infinity, $\lambda=0$ is in a spectral
gap of $-\Delta+V$, and $\partial_t F$ denotes the generalized gradient of $F$
with respect to variable $t$. Here, by employing Variational Methods for
Locally Lipschitz Functionals, we establish the existence of solution when $f$
is periodic and non periodic
