Ground states for a fractional scalar field problem with critical growth

Abstract

We prove the existence of a ground state solution for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u= g(u) \mbox{ in } \mathbb R^{N} \end{align*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, and $g\in C^{1, \beta}( \mathbb R, \mathbb R)$ is an odd function satisfying the critical growth assumption.