  We consider the following class of fractional Schr\"odinger equations $$
(-\Delta)^{\alpha} u + V(x)u = K(x) f(u) \mbox{in} \mathbb{R}^{N} $$ where
$\alpha\in (0, 1)$, $N>2\alpha$, $(-\Delta)^{\alpha}$ is the fractional
Laplacian, $V$ and $K$ are positive continuous functions which vanish at
infinity, and $f$ is a continuous function. By using a minimization argument
and a quantitative deformation lemma, we obtain the existence of a
sign-changing solution. Furthermore, when $f$ is odd, we prove that the above
problem admits infinitely many nontrivial solutions. Our result extends to the
fractional framework some well-known theorems proved for elliptic equations in
the classical setting. With respect to these cases studied in the literature,
the nonlocal one considered here presents some additional difficulties, such as
the lack of decompositions involving positive and negative parts, and the
non-differentiability of the Nehari Manifold, so that a careful analysis of the
fractional spaces involved is necessary.
