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.
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机译：我们研究了Schr \“ {o} dinger方程：\ begin {eqnarray}-\ Deltau + V（x）u + f（x，u）= 0，\ qquad u \ in H ^ {1}（\ mathbb { R} ^ {N}），\ nonumber \ end {eqnarray}，其中$ V $是周期的，而$ f $是周期的，在$ x $变量中，$ 0 $在运算符$-\ Delta的范围内+ V $我们证明，在$ f $的一些新假设下，该方程具有非平凡的解，我们对非线性$ f $的假设非常弱，并且与文献中的已知假设有很大不同。
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