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

TL;DR

This paper proves the existence of solutions for a Schr"odinger equation with a periodic potential and non-periodic nonlinearity, using advanced variational methods to extend previous results in the field.

Contribution

It introduces a new existence result for Schr"odinger equations with periodic potentials and non-periodic nonlinearities, employing linking theorem and spectral analysis techniques.

Findings

Established existence of solutions in weak superlinear cases

Extended previous results by Willem and Zou

Applied spectral and variational methods to non-periodic nonlinearities

Abstract

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].