  In this paper, we study the following nonlinear problem of Kirchhoff type
with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)\Delta
u+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where $a,$ $b>0$ are
constants, $2<p<5$ and $V:\R^3\rightarrow\R$. Under certain assumptions on $V$,
we prove that \eqref{0.1} has a positive ground state solution by using a
monotonicity trick and a new version of global compactness lemma. Our main
results can be viewed as a partial extension of a recent result of He and Zou
in [16] concerning the existence of positive solutions to the nonlinear
Kirchhoff problem (\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D
u|^2\right)\Delta u+V(x)u=f(u), u\in H^1(\R^3), u>0, $x\in \R^3$, where
$\varepsilon>0$ is a parameter, $V(x)$ is a positive continuous potential and
$f(u)\thicksim |u|^{p-1}u$ with $3<p<5$ and satisfies the Ambrosetti-Rabinowitz
type condition. Our main results extend also the arguments used in [7,36],
which deal with Schr\"{o}dinger-Poisson system with pure power nonlinearities,
to the Kirchhoff type problem.
