In this article we prove existence of positive solutions for the nonlinear elliptic equation M_(λ,Λ)~+(D~2u) - γu + f(u) = 0 in Ω, u = 0 on (partial deriv)Ω, where M_(λ,Λ)~+ denotes Pucci's extremal operator with parameters 0 < λ ≤ Λ and Ω is convex smooth domain in R~N, N ≥ 3. The result applies to a class of nonlinear functions f, including the model cases: i) γ = 1 and f(s) = s~p, 1 < p ≤ p~+; and ii) γ = 0, f(s) = αs + s~p, 1 < p ≤ p~+, and 0 < α < μ_1~+. Here p~+ = (N-bar)~+/{(N-bar)~+ - 2), N~+ = λ(N - 1)/Λ + 1, and μ_1~+ is the first, eigenvalue of M_(λ,Λ)~+ in Ω. Analogous results are obtained for the operator M_(λ,Λ)~-.
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