  In this paper, the existence of least energy solution and infinitely many
solutions is proved for the equation $(1-\Delta)^\alpha u = f(u)$ in
$\mathbf{R}^N$ where $0<\alpha<1$, $N \geq 2$ and $f(s)$ is a Berestycki-Lions
type nonlinearity. The characterization of the least energy by the mountain
pass value is also considered and the existence of optimal path is shown.
Finally, exploiting these results, the existence of positive solution for the
equation $(1-\Delta)^\alpha u = f(x,u)$ in $\mathbf{R}^N$ is established under
suitable conditions on $f(x,s)$.
