  We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in
a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or
Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and
changes sign, in which case the strong maximum principle does not apply. As a
consequence, the set of nonnegative solutions of these problems has a rich
structure, featuring in particular both dead core and/or positive solutions.
Overall, we are interested in sufficient and necessary conditions on $a$ and
$q$ for the existence of positive solutions. We describe the main results from
the past decades, and combine it with our recent contributions. The proofs are
briefly sketched.
