On Schrödinger equations with periodic potentials and with nonperiodic nonlinearities

We consider the Schrödinger equation $-Δu +V(x)u=f(x, u)$, where $V$ is periodic and $f$ is non-periodic, 0 is a boundary point of the continuous spectrum of $A:=-Δ+V(x)$. 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 Schrödinger Equation with Periodic Potential and Spectrum Point Zero,Indiana Univ. Math. J. 2003].

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