  Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a
possibly discontinuous and unbounded function that changes sign in $\Omega$.
Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a
continuous function such that $k_{1}\xi^{p}\leq f\left(\xi\right) \leq
k_{2}\xi^{p}$ for all $\xi\geq0$ and some $k_{1},k_{2}>0$ and
$p\in\left(0,1\right) $. We study existence and nonexistence of strictly
positive solutions for nonlinear elliptic problems of the form $-\Delta
u=m\left(x\right) f\left(u\right) $ in $\Omega$, $u=0$ on $\partial\Omega$.
