Abstract: This paper is concerned with the existence of positive solutions of the following Dirichlet problem for p-mean curvature operator with critical exponent: ... is a bounded domain in R^N(N＞p＞1)with smooth boundary Ω,2＜=p＜=q＜=P*,P*=(Np)/(N-p),λ,P＞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. 

Key words: mean curvature operator, critical exponent, (PS)condition, dual set