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In this paper we deal with the following fractional Kirchhoff equation\begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-\Delta)^{s}u = g(u) \mbox{ in }\mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is asmall positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an oddfunction satisfying Berestycki-Lions type assumptions. By using minimaxarguments, we establish a multiplicity result for the above equation, providedthat $q$ is sufficiently small.

A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity