Let Ω ⊂ R N be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) u p − b (x, t) u q in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the existence of solutions u λ in the interior of the positive cone and that u λ can be chosen such

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... that λ → u λ is differentiable and increasing. A uniqueness theorem is also given in the case p ≤ q. All results remain valid for the corresponding elliptic problems. 2000 Mathematics Subject Classification. 35K20, 35P05, 35B10.