  In the present paper, we establish the uniqueness and nondegeneracy of
positive energy solutions to the Kirchhoff equation \begin{eqnarray*}
-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+u=|u|^{p-1}u & &
\text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants.
Then, as applications, we derive the existence and local uniqueness of
solutions to the perturbed Kirchhoff problem
  \begin{eqnarray*} -\left(\epsilon^2a+\epsilon b\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\right)\Delta u+V(x)u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}
\end{eqnarray*} for $\epsilon>0$ sufficiently small, under some mild
assumptions on the potential function $V:\mathbb{R}^3\to \mathbb{R}$. The
existence result is obtained by applying the Lyapunov-Schmidt reduction method.
It seems to be the first time to study singularly perturbed Kirchhoff problems
by reduction method, as all the previous results were obtained by various
variational methods. Another advantage of this approach is that it gives a
unified proof to the perturbation problem for all $p\in (1,5)$, which is quite
different from using variational methods in the literature. The local
uniqueness result is totally new. It is obtained by using a type of local
Pohozaev identity, which is developed quite recently by Deng, Lin and Yan in
their work "On the prescribed scalar curvature problem in $\mathbb{R}^N$, local
uniqueness and periodicity." (see J. Math. Pures Appl. (9) {\bf 104}(2015),
1013-1044).
