This paper is concerned with the existence and nodal character of the nontrivial solutions for the following quasilinear elliptic equations involving critical Sobolev exponents: (1)- i=1N x i | u| p - 2 u x i +|u| p - 2 u=|u| p * - 2 u+f(u),u W r 1 , p (R N ),where p>=2 and p * =Np/(N-p) is the critical Sobolev exponent for the embedding W r 1 , p (R N ) L p * (R N ). The function f satisfies some conditions given by (f 1 ),(f 2 ),(f 3 ) in the paper. The main results obtained in this paper are that there exists at least a pair of nontrivial solutions for provided that N>=p 2 and there exists at least a pair of nontrivial solutions u k + , u k - of for each k N {0} such that both u k + , and u k - possess exactly k nodes provided that (f(t))''>=0 for t>=0 and N>=p 2 (p-1)+p.