  The present paper deals with a parametrized Kirchhoff type problem involving
a critical nonlinearity in high dimension. Existence, non existence and
multiplicity of solutions are obtained under the effect of a subcritical
perturbation by combining variational properties with a careful analysis of the
fiber maps of the energy functional associated to the problem. The particular
case of a pure power perturbation is also addressed. Through the study of the
Nehari manifolds we extend the general case to a wider range of the parameters.
