Existence of nontrivial solutions for asymptotically linear periodic Schr\"odinger equations

We study the Schr\"{o}dinger equation: \begin{equation} - \Delta u+V(x)u=f(x,u) ,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{equation} where $V$ is periodic and $f$ is periodic in the $x$-variables, 0 is in a gap of the spectrum of the operator $-\Delta+V$ and $f$ is asymptotically linear as $|u|\rightarrow+\infty.$ We prove that under some asymptotically linear assumptions for $f$, this equation has a nontrivial solution. Our assumptions for $f$ are different from the classical assumptions raised by G. B. Li and A. Szulkin in