Concentration phenomena for a class of fractional Kirchhoff equations in ${\mathbb{R}}^N$ with general nonlinearities

In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-\Delta)^{s}u + V(x) u= f(u) &\mbox{ in } $\mathbb{R}$^{N}, \ u\in H^{s}($\mathbb{R}$^{N}), \quad u>0 &\mbox{ in } $\mathbb{R}$^{N}, \end{array} \right. \end{equation*} where $\epsilon >0$ is a small parameter, $s\in (0,1)$, $N\ge 2$, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a positive continuous function, $M:[0,\mathrm{\infty })\to \mathbb{R}$ is a Kirchhoff function satisfying suitable conditions and $f:$\mathbb{R}$\rightarrow $\mathbb{R}$$ fulfills Berestycki-Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions $(u_{\epsilon })$ which concentrates at a local minimum of $V$ as $\epsilon \to 0$.