  We study the concentration and multiplicity of weak solutions to the
Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered}
  - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{{\R^3}} {{{\left| {\nabla u}
\right|}^2}} \Bigr)\Delta u + V(z)u
  = f(u) + {u^5}{\text{in}}{\R^3}, \hfill u \in {H^1}({\R^3}),{\text{}}u >
0{\text{in}}{\R^3}, \hfill \\ \end{gathered} \right.\] where $\varepsilon $ is
a small positive parameter and $a,b > 0$ are constants, $f \in {C^1}({\R^
+},\R)$ is subcritical, $V:{\R^3} \to \R$ is a locally H\"{o}lder continuous
function.
  We first prove that for ${\varepsilon_0} > 0$ sufficiently small, the above
problem has a weak solution ${u_\varepsilon}$ with exponential decay at
infinity. Moreover, ${u_\varepsilon}$ concentrates around a local minimum point
of $V$ in $\Lambda $ as $\varepsilon \to 0$. With minimax theorems and
Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing
the topology construct of the set where the potential $V\left(z \right)$
attains its minimum.
