  Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and
let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We
investigate the existence of positive solutions for the semipositone problem
$-\Delta u=\lambda m(x)(f(u)-k)$ in $\Omega$, $u=0$ on $\partial\Omega$, where
$\lambda,k>0$ 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 $\lambda$ provided that the negative part of $m$ is suitably small.
Our main tool is the sub-supersolutions method, combined with some rescaling
properties.
