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

Abstract

In this paper we study existence of ground state solution to the following problem $$ (- Δ)^αu = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^α(\mathbb R^N) $$ where $(-\Delta {)}^{\alpha }$ is the fractional Laplacian, $\alpha \in (0,1)$. We treat both cases $N\ge 2$ 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 $N\ge 2$, exponential if $N=1$.