We are concerned with the existence and concentrating phenomenon of positive ground state solutions for the following class of Schrödinger-Poisson systems involving competing potentials and doubly critical growth \( \left\{ \begin{array}{ll} -\epsilon ^2\Delta u+ V(x)u-K(x)\phi |u|^3u=K(x)|u|^4u+ Q(x)f(u), & x\in {\mathbb {R}}^3, \\ -\epsilon ^2\Delta \phi =K(x)|u|^5, & x\in {\mathbb {R}}^3,\\ \end{array}\right. \) where \(\epsilon >0\) is a small parameter. Under some suitable assumptions on V, Q, K and f, we deduce that this system admits a positive ground state solution for all sufficiently small \(\epsilon >0\) by using variational methods, where the decaying 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 additionally investigate the nonexistence of ground state solutions.
