Ground state solutions for fractional scalar field equations under a general critical nonlinearity

In this paper we study existence of ground state solution to the following problem $$(-\mathrm{\Delta }{)}^{\alpha }u=g(u)in{\mathbb{R}}^N,u\in H^{\alpha }({\mathbb{R}}^N)$$ where $(-\mathrm{\Delta }{)}^{\alpha }$ is the fractional Laplacian, $\alpha \in (0,1)$. We treat both cases $N\ge 2$ and $N=1$ with $\alpha =1\mathrm{/}2$. The function $g$ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $N\ge 2$, exponential if $N=1$.