  We consider the Schr\"{o}dinger equation $-\Delta u +V(x)u=f(x, u)$, where
$V$ is periodic and $f$ is non-periodic, 0 is a boundary point of the
continuous spectrum of $A:=-\Delta +V(x)$. We use M. Willem and W. M. Zou's
linking theorem and M. Schechter's method to establish an existence result for
this problem in weak superlinear cases. In a sense, we enrich a recent result
of M. Willem and W.M. Zou [M. Willem and W.M. Zou, On a Schr\"{o}dinger
Equation with Periodic Potential and Spectrum Point Zero,Indiana Univ. Math. J.
2003].
