Existence of nontrivial solutions for periodic Schrodinger equations with new nonlinearities

Abstract

We study the Schrödinger equation: \begin{eqnarray} - Δu+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of the operator $-\Delta +V$. 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.

arXiv admin note: substantial text overlap with arXiv:1310.2399