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We study the Schr\"{o}dinger equation: \begin{eqnarray} - \Deltau+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 gapof the spectrum of the operator $-\Delta+V$. We prove that under some newassumptions for $f$, this equation has a nontrivial solution. Our assumptionsfor the nonlinearity $f$ are very weak and greatly different from the knownassumptions in the literature.

Existence of nontrivial solutions for periodic Schrodinger equations with new nonlinearities