In this work we present some partial results that will appear in a completed from in a forthcoming paper, [7]. We discuss the existence and particularly the multiplicity of solutions for the nonlinear system of elliptic equations. Δui + λfi (x, u1, . . . um) = 0    in    Ω      (1.1) uiǀ𝜃Ω = 0,    i = 1 , . . . , m                     (1.2) Where fi (x, 0, . . . 0) > 0 for all x ⋲ Ω,  i = 1, 2, . . . , m. The functions fi,  i = 1, . . . , m, satisfy the quasimonotone condition and a certain blow up rate us to be made precise in the assumptions (H1) and (H2) below. Then results similar to those of the scalar equation case (see [6]) can be established.