Abstract: In this paper is given the existence theorem of n on-trivial solution for a class of semilinear elliptic equations with the criti cal Sobolev exponent considering the first characteristic value by means of the Mountain Pass Lemma without the condition (PS), and of the P L Lions' concentrat ion-compactness principle.

Key words: semilinear elliptic equation, critial Sobolev exponent, dirichlet problem, eigen-value, concentration-compactness principle