  In the present paper, we consider the nonlocal Kirchhoff problem
\begin{eqnarray*} -\left(\epsilon^2a+\epsilon b\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\right)\Delta u+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3},
\end{eqnarray*} where $a,b>0$, $1<p<5$ are constants, $\epsilon>0$ is a
parameter. Under some mild assumptions on the function $V$, we obtain
multi-peak solutions for $\epsilon$ sufficiently small by Lyapunov-Schmidt
reduction method. Even though many results on single peak solutions to
singularly perturbed Kirchhoff problems have been derived in the literature by
various methods, there exist no results on multi-peak solutions before this
paper, due to some difficulties caused by the nonlocal term
$\left(\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u$. A remarkable new
feature of this problem is that the corresponding unperturbed problem turns out
to be a system of partial differential equations, but not a single Kirchhoff
equation, which is quite different from most of elliptic singular perturbation
problems.
