  In this paper, a class of Schr\"{o}dinger-Poisson system involving multiple
competing potentials and critical Sobolev exponent is considered. Such a
problem cannot be studied with the same argument of the nonlinear term with
only a positive potential, because the weight potentials set $\{Q_i(x)|1\le i
\le m\}$ contains nonpositive, sign-changing, and nonnegative elements. By
introducing the ground energy function and subtle analysis, we first prove the
existence of ground state solution $v_\varepsilon$ in the semiclassical limit
via the Nehari manifold and concentration-compactness principle. Then we show
that $v_\varepsilon$ converges to the ground state solution of the associated
limiting problem and concentrates at a concrete set characterized by the
potentials. At the same time, some properties for the ground state solution are
also studied. Moreover, a sufficient condition for the nonexistence of the
ground state solution is obtained.
