  This paper is devoted to study a class of nonlinear fractional
Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad
\text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$,
$(-\Delta)^{s}$ stands for the fractional Laplacian. First, by using a
variational approach, we establish the existence of at least one nontrivial
solution for the above equation with a general potential $V(x)$ which is
allowed to be sign-changing and a sublinear nonlinearity $f(x,u)$. Next, by
using variational methods and the Moser iteration technique, we prove the
existence of infinitely many solutions with $V(x)$ is a nonnegative potential
and the nonlinearity $f(x,u)$ is locally sublinear with respect to $u$.
