Ground state solutions for a fractional Schr\"odinger equation with critical growth

In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u+V(x)u= f(u) \mbox{ in } $\mathbb{R}$^{N}, \end{align*} where $s\in (0,1)$, $N>2s$, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $V: $\mathbb{R}$^{N}\rightarrow $\mathbb{R}$$ is a bounded potential satisfying suitable assumptions, and $f\in C^{1,\beta }(\mathbb{R},\mathbb{R})$ has critical growth. We first analyze the case $V$ constant, and then we develop a Jeanjean-Tanaka argument \cite{JT} to deal with the non autonomous case. As far as we know, all results presented here are new.