Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

In this paper we consider positive supersolutions of the nonlinear elliptic equation \begin{document}$ - \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega, $\end{document} where \begin{document}$ 0\le p , \begin{document}$ \Omega $\end{document} is an arbitrary domain (bounded or unbounded) in \begin{document}$ {\mathbb{R}}^N $\end{document} ( \begin{document}$ N\ge 2 $\end{document} ), \begin{document}$ f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+} $\end{document} \begin{document}$ (0 is a non-decreasing continuous function and \begin{document}$ \rho: \Omega \rightarrow \mathbb{R} $\end{document} is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions \begin{document}$ u $\end{document} at each point \begin{document}$ x\in\Omega $\end{document} where \begin{document}$ \nabla u\not\equiv0 $\end{document} in a neighborhood of \begin{document}$ x $\end{document} . As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains \begin{document}$ \Omega $\end{document} with the property that \begin{document}$ \sup_{x\in\Omega}dist (x, \partial\Omega) = \infty $\end{document} . In particular when \begin{document}$ \rho(x) = |x|^\beta $\end{document} ( \begin{document}$ \beta\in {\mathbb{R}} $\end{document} ) and \begin{document}$ f(u) = u^q $\end{document} with \begin{document}$ q+p>1 $\end{document} then every positive supersolution in an exterior domain is eventually constant if \begin{document}$ (N-2)q+p(N-1)