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We study the semilinear Schrödinger equation
$$\left\{ _{u\,\,\in \,\,{{H}^{1}}({{\mathbf{R}}^{N}}),}^{-\Delta \,u+V(x)u=f(x,u),\,\,\,\,\,x\in \,\,{{\mathbf{R}}^{N}},} \right.$$
where 
$f$ is a superlinear, subcritical nonlinearity. It focuses on the case where 
$V(x)={{V}_{0}}(x)+{{V}_{1}}(x)$, 
${{V}_{0}}\in C({{\mathbf{R}}^{N}}),\,{{V}_{0}}(x)$ is 1-periodic in each of 
${{x}_{1}},{{x}_{2}},...,{{x}_{N}}$, 
$\sup [\sigma (-\Delta +{{V}_{0}})\,\cap \,(-\infty ,0)]<0<$
$\inf [\sigma (-\Delta +{{V}_{0}})\cap (0,\infty )],\,{{V}_{1}}\in C({{\mathbf{R}}^{N}})$, and 
${{\lim }_{|x|\to \infty }}\,{{V}_{1}}(x)=0$. A new super-quadratic condition is obtained that is weaker than some well-known results.
