  We consider a Kirchhoff problem of Brezis-Nirenberg type in a smooth bounded
domain of $\mathbb{R}^4$ with Dirichlet boundary conditions. Our approach,
novel in this framework and based upon approximation arguments, allows us to
cope with the interaction between the higher order Kirchhoff term and the
critical nonlinearity, typical of the dimension four. We derive several
existence results of positive solutions, complementing and improving earlier
results in the literature. In particular, we provide explicit bounds of the
parameters $b$ and $\lambda$ coupled, respectively, with the higher order
Kirchhoff term and the subcritical nonlinearity, for which the existence of
solutions occurs.
