We are concerned with the existence and concentrating behavior of positive ground state solutions for a quasilinear Kirchhoff equation involving critical Sobolev exponent with competing potentials \( \begin{gathered} \left( \epsilon ^2a+\epsilon b\int _{\mathbb {R}^3}g^2(u)|\nabla u|^2\textrm{d}x\right) \left[ -\text {div}(g^2(u)\nabla u)+ g(u)g^{\prime }(u)|\nabla u|^2\right] +V( x)u\\ \ \ \ \ \ \ =Q( x)h(u)+K( x)|G(u)|^{4}G(u)g(u),~x\in \mathbb {R}^3,\\ \end{gathered} \) where \(a,b>0\) are constants, \(\epsilon >0\) is a small parameter, and g is an even differential function related to the quasilinear term, such that \(G(t)=\int _0^tg(s)\textrm{d}s\) . Under some suitable assumptions on V, Q, K and h, we conclude that this equation admits a positive ground state solution for all sufficiently small \(\epsilon >0\) using variational methods, where the decay rate of the obtained solution as \(|x|\rightarrow +\infty \) and its concentration on the set of minimal points of V and the sets of maximal points of Q and K as \(\epsilon \rightarrow 0^+\) are also considered. In particular, we also investigate the nonexistence of ground state solutions.
