  We investigate the existence of least energy solutions and infinitely many
solutions for the following nonlinear fractional equation (-\Delta)^{s} u =
g(u) \mbox{ in } \mathbb{R}^{N}, where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$
is the fractional Laplacian and $g: \mathbb{R} \rightarrow \mathbb{R}$ is an
odd $\mathcal{C}^{1, \alpha}$ function satisfying Berestycki-Lions type
assumptions. The proof is based on the symmetric mountain pass approach
developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the
mountain pass approach and an approximation argument, we also prove the
existence of a positive radially symmetric solution for the above problem when
$g$ satisfies suitable growth conditions which make our problem fall in the so
called "zero mass" case.
