  In this paper we deal with a stationary non-degenerate $p-$Kirchhoff type
problem with critical non-linearity and a subcritical parametrized
perturbation. We work on bounded domains of the Euclidean space, without any
restriction on the dimension or on $p>0$. Variational methods will be utilized
in combination with an analysis of the fibering functions of the energy
functional, in order to obtain ground state solutions, as well as Mountain Pass
solutions, depending on the values of the parameter. A local analysis of the
energy functional will allow us to obtain non-trivial solutions even beyond the
extremal parameter.
