On Schr\"odinger equations with periodic potentials and with nonperiodic nonlinearities

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].