Mountain pass solutions for the fractional Berestycki-Lions problem

We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation egin{equation*} (-Delta)^{s}u = g(u) mbox{ in } mathbb{R}^{N} end{equation*} where $sin (0,1)$, $Ngeq 2$, $(-Delta)^{s}$ is the fractional Laplacian and $g: mathbb{R} ightarrow mathbb{R}$ is an odd $C^{1, alpha}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in [33]. Moreover, by combining the moun- tain 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.