Existence of solutions of scalar field equations with fractional operator

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 $f(x,s)$.