  We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$,
of positive solutions for the problem $(P_{a,q})$: $-\Delta u=a(x)u^{q}$ in
$\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded and smooth
domain of $\mathbb{R}^{N}$ and $a:\Omega\rightarrow\mathbb{R}$ is a
sign-changing function (in which case the strong maximum principle does not
hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as
$q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases
$u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain
existence results for a singular and indefinite Dirichlet problem. Our results
are mainly based on bifurcation and sub-supersolutions methods.
