Abstract: In this paper, a necessary condition on the existence of a non-trivia l solution for the semi-linear elliptic equation (-Δu=λ₁u+|u|^{2 *-2}u) with zero boundary, the critical Sobolev exponents and the eigenvablue (λ₁) is given, so is obtained the correspondent co nclusion in weighted condition.

Key words: semi-linear elliptic equation, critical Sobolev exponent, first eigenvalue, weight function