  In this paper we are concerned with the fractional Schr\"{o}dinger equation
$(-\Delta)^{\alpha} u+V(x)u =f(x, u)$, $x\in \rn$, where $f$ is superlinear,
subcritical growth and $u\mapsto\frac{f(x, u)}{\vert u\vert}$ is nondecreasing.
When $V$ and $f$ are periodic in $x_{1},\ldots, x_{N}$, we show the existence
of ground states and the infinitely many solutions if $f$ is odd in $u$. When
$V$ is coercive or $V$ has a bounded potential well and $f(x, u)=f(u)$, the
ground states are obtained. When $V$ and $f$ are asymptotically periodic in
$x$, we also obtain the ground states solutions. In the previous research,
$u\mapsto\frac{f(x, u)}{\vert u\vert}$ was assumed to be strictly increasing,
due to this small change, we are forced to go beyond methods of smooth
analysis.
