  We study the following Brezis-Nirenberg problem of Kirchhoff type $$
\left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u
+ \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\
\partial\Omega, \endaligned \right. $$ where $\Omega\subset \bbr^4$ is a
bounded domain with the smooth boundary $\partial\Omega$, $2\leq q<4$ and $a$,
$b$, $\lambda$, $\delta$ are positive parameters. We obtain some new existence
and nonexistence results, depending on the values of the above parameters,
which improves some known results. The asymptotical behaviors of the solutions
are also considered in this paper.
