We study a class of degenerate Kirchhoff-type equations driven by the p-Laplacian and involving critical Sobolev growth and subcritical perturbations. The problem is set in a bounded domain \(\Omega \subset {\mathbb {R}}^N\) with \(N \ge p^2\) , and takes the form \(\begin{aligned} & -\left( a - b\left( \int _\Omega |\nabla u|^p\, dx\right) ^{k-1}\right) \Delta _p u \\ & \quad = \lambda |u|^{q-2} u + K(x) |u|^{p^*-2} u \quad \text {in } \Omega , \quad u = 0 \text { on } \partial \Omega , \end{aligned}\) where \( 1 < q \le p \) , \( k = \frac{p^{*}}{p} \) , and \( p^{*} = \frac{Np}{N-p} \) . We establish the existence of ground state solutions using variational methods and a refined mountain pass framework. A key analytical difficulty arises from the combination of the nonlocal Kirchhoff term and the lack of compactness due to the critical exponent. We overcome this by deriving sharp energy estimates and verifying the Palais–Smale condition. Furthermore, we provide a detailed asymptotic analysis of solution profiles as \(\lambda \) approaches a critical threshold, highlighting the effect of degeneracy in K(x).
