  Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain
and $a:\Omega\rightarrow\mathbb{R}$ be a sign-changing weight satisfying
$\int_{\Omega}a<0$. We prove the existence of a positive solution $u_{q}$ for
the problem $(P_{a,q})$:
  $-\Delta u=a(x)u^{q}$ in $\Omega$, $\frac{\partial u}{\partial\nu}=0$ on
$\partial\Omega$,
  if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the
existence result previously established in [16]. In addition, we provide the
asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $\Omega$ is a ball
and $a$ is radial, we give some explicit conditions on $q$ and $a$ ensuring the
existence of a positive solution of $(P_{a,q})$. We also obtain some properties
of the set of $q$'s such that $(P_{a,q})$ admits a solution which is positive
on $\overline{\Omega}$. Finally, we present some results on nonnegative
solutions having dead cores. Our approach combines bifurcation techniques, a
priori bounds and the sub-supersolution method. Several methods and results
apply as well to the Dirichlet counterpart of $(P_{a,q})$.
