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 $$(- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N)$$ where $(-\Delta)^{\alpha}$ is the fractional Laplacian, $\alpha\in (0,1)$. We treat both cases $N\geq2$ and $N=1$ with $\alpha=1/2$. The function $g$ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $N\geq2$, exponential if $N=1$.