This paper is concerned with the existence of positive solutions of the following Dirichlet problem for p-mean curvature operator with critical exponent:-div((1+|(△)u|2)p-2/2(△)u)=λup*-1+μuq-1,u = 0, x ∈ (a)Ω,where u ∈ W1,p0(Ω),Ω is a bounded domain in RN(N ＞ p ＞ 1) with smooth boundary (a)Ω, 2 ≤ p ≤ q ＜ p*, p* = Np/N-p,λ, μ＞0. It reaches the conclusions that this problem has at least one positive solution in the different cases. It is discussed the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with critical exponent by using Nehari-type duality property firstly. As p = 2, q = p, the result is correspond to that of Laplace operator.