  This paper is dedicated to studying the semilinear Schr\"odinger equation
$\left\{\begin{array}{ll}-\Nabla u+V(x)u=f(x, u), \ \ \ \ x\in {\R}^{N},u\in
H^{1}({\R}^{N}),\end{array}\right.$ where $f$ is a superlinear, subcritical
nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in
C(\R^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and
$\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]<0<\inf[\sigma(-\triangle
+V_0)\cap (0, \infty)]$, $V_1\in C(\R^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A
new super-quadratic condition is obtained, which is weaker than some well known
results.
