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 $s\in (0,1)$, $N> 2s$, $(-\Delta)^{ s}$ 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.
Categories: math.AP