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We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth  where ε is a small positive parameter and a, b > 0 are constants, f ∈ C 1 (R + , R) is subcritical, V : R 3 → R is a locally Hölder continuous function. We first prove that for ε 0 > 0 sufficiently small, the above problem has a weak solution u ε with exponential decay at infinity. Moreover, u ε concentrates around a local minimum point of V in Λ as ε → 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 (z) attains its minimum.

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