Let Ω ⊂ RN, N ≥ 1, be a smooth bounded domain, and let m : Ω → R be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem {=Δu = λm(x)(f(u) – k) in Ω u = 0 on ω where λ; k > 0 and f is either sublinear at infinity with f(0) = 0, or f has a singularity at 0. We prove the existence of a positive solution for certain ranges of λ provided that the negative part of m is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.
