On Schrödinger equations with periodic potentials and with nonperiodic nonlinearities

Abstract

We consider the Schrödinger equation $-\hat{I}”u+V(x)u=f(x,u)$, where $V$ is periodic and $f$ is non-periodic, 0 is a boundary point of the continuous spectrum of $A:=-\hat{I}”+V(x)$. 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].

This paper has been withdrawn by the author due to a crucial sign error in equation