The existence and concentration of ground-state solutions for a class of Kirchhoff type problems in R3${\mathbb{R}^{3}}$ involving critical Sobolev exponents

where ε is a small positive parameter, a,b > 0, λ > 0, 2 < p≤ 4, V andW are two potentials. Under proper assumptions, we prove that, for ε > 0 sufficiently small, the above problem has a positive ground-state solution uε by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to Gui (Commun. Partial Differ. Equ. 21:787-820, 1996) to show that uε is concentrated around a set which is related to the set where the potential V(x) attains its global minima or the set where the potentialW(x) attains its global maxima as ε→ 0.