  We consider the semilinear Schr\"odinger equation
  $$
  \left\{
  \begin{array}{ll}
  -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\R}^{N},
  u\in H^{1}({\R}^{N}),
  \end{array}
  \right.
  $$
  where $f$ is a superlinear, subcritical nonlinearity. We mainly study the
case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbb{R}^N)$, $V_0(x)$ is
1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle
+V_0)\cap (-\infty, 0)]<0<\inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$,
$V_1\in C(\mathbb{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. Inspired by
previous work of Li et al. \cite{LWZ}, Pankov \cite{Pa} and Szulkin and Weth
\cite{Sz}, we develop a more direct approach to generalize the main result in
\cite{Sz} by removing the "strictly increasing" condition in the Nehari type
assumption on $f(x, t)/|t|$. Unlike the Nahari manifold method, the main idea
of our approach lies on finding a minimizing Cerami sequence for the energy
functional outside the Nehari-Pankov manifold $\mathcal{N}^{0}$ by using the
diagonal method.
