By using the mountain pass lemma, we study the existence of positive solutions for the equation − Δ u ( x ) = λ ( u | u | + u ) ( x ) for x ∈ Ω together with Dirichlet boundary conditions and show that for every λ < λ 1 , where λ 1 is the first eigenvalue of − Δ u = λ u in Ω with the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists for λ ≥ λ 1 .