The article aims to study new and current problems in the theory of nonclassical partial differential equations and their applications, proving the existence and nonexistence of solutions to $p$-Kirchhoff type problems with {critical exponent of Sobolev} in $\mathbb{R}^{n}$, which are of great interest in the study of mathematical physics equations. We show the existence of a local minimizer with negative/positive energy by using variational methods. More precisely, we considered a minimization of $ E_{\lambda }$ constrained in a neighborhood of zero using the Ekeland variational principle, then we found the first critical point of $E_{\lambda }$ which achieves the local minimum of $E_{\lambda }$ whose level is negative; next around the zero point, using the mountain pass theorem, we also obtained a critical point whose level is positive. In addition, we studied the case of $\lambda =0$, where there is no non-trivial solution using the contradiction principle. We also {established infinite solutions} and discuss the different cases.