Three positive solutions for semilinear elliptic problems involving concave and convex nonlinearities

We study the existence and multiplicity of positive solutions for the Dirichlet problem -Δu=λf(z)|u| ^q-2u+|u| ^p-2u in Ω, where λ > 0, 1 < q < 2, p = 2* = 2N/(N - 2), 0 ∈ Ω ⊂ ℝ ^N, N ≥ 3, is a bounded domain with smooth boundary ∂Ω and f is a non-negative continuous function on Ω̄. Assuming that f satisfies some hypothesis, we prove that the equation admits at least three positive solutions for sufficiently small λ.