Abstract: We consider the nonlinear elliptic equation with mixed boundary conditions
(P)-△u=b(x)u^p+λu,u>0 for x∈Ω,u|_Γ0=0,(∂u)/∂n|_Γ1=0, where Ω is a smooth bounded domain in R^N (N ≥ 3), whose boundary ∂Ω is made of two open sets Γ₀ and Γ₁,Γ₀∩Γ₁=ϕ, ∂Ω=Γ₀∪Γ₁,p=(N+2)/N-2. We prove the existence of a positive solution of (P)when b (x) and λ satisfy suitable conditions.