  In this paper, by using variational methods we study the existence of
positive solutions for the following Kirchhoff type problem: $$ \left\{
\begin{array}{ll} -\left(a+b\mathlarger{\int}_{\Omega}|\nabla
u|^{2}dx\right)\Delta u+V(x)u=u^{5}, \ & x\in\Omega,\\ \\ u=0,\ & x\in\partial
\Omega, \end{array}\right. $$ where $a>0$, $b\geq0$, $\Omega\subset\mathbb R^3$
is an unbounded exterior domain, $\partial\Omega\neq\emptyset$,
$\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in D_{0}^{1,2}(\Omega)$, and
$V\in L^{\frac{3}{2}}(\Omega)$ is a non-negative continuous function. It turns
out that the above Kirchhoff equation has no ground state solution.
Nonetheless, by establishing some global compact lemma and constructing a
suitable minimax value $c$ at a higher energy level where so called
Palais-Smale condition holds, we succeed to obtain a positive solution for such
a problem whenever $V$ and the hole $\mathbb{R}^{3}\setminus\Omega$ are
suitable small in some senses. To the best of our knowledge, there are few
similar results published in the literature concerning the existence of
positive solutions for Kirchhoff equation in exterior domains. Our result also
holds true in the case $\Omega=\mathbb R^3$, particularly, if $a=1$ and $b=0$,
we improve some existing results (such as Benci, Cerami, Existence of positive
solutions of the equation $-\Delta u+a(x)u=u^{(N+2)/(N-2)}$ in $\emph{R}^{N}$,
J. Funct. Anal., 88 (1990), 90--117) for the corresponding Schr\"odinger
equation in the whole space.
