Existence of Nontrivial Solutions for Periodic Schrödinger Equations with New Nonlinearities

Abstract

We study the Schrödinger equation: $-\mathrm{\Delta }u+V\left(x\right)u+f\left(x,u\right)=0,\hspace{0.17em}\hspace{0.17em}u\in H^1({\mathbb{R}}^N)$, where $V$ is $1$-periodic and $f$ is $1$-periodic in the $x$-variables; $0$ is in a gap of the spectrum of the operator $-\mathrm{\Delta }+V$. We prove that, under some new assumptions for $f$, this equation has a nontrivial solution. Our assumptions for the nonlinearity $f$ are very weak and greatly different from the known assumptions in the literature.