  In this paper we consider positive supersolutions of the nonlinear elliptic
equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in }
\Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or
unbounded) in $ \IR^N$ ($N\ge 2$), $f: [0,a_{f}) \rightarrow \Bbb{R}_{+}$ $(0 <
a_{f} \leqslant +\infty)$ is a non-decreasing continuous function and $\rho:
\Omega \rightarrow \IR$ is a positive function. Using the maximum principle we
give explicit estimates on positive supersolutions $u$ at each point
$x\in\Omega$ where $\nabla u\not\equiv0$ in a neighborhood of $x$. As
consequences, we discuss the dead core set of supersolutions on bounded
domains, and also obtain Liouville type results in unbounded domains $\Omega$
with the property that $\sup_{x\in\Omega}dist (x,\partial\Omega)=\infty$.
