In this paper we are concerned with the fractional Schrödinger equation $ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $, $ x\in {{\mathbb{R}}^{N}} $, 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.
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