Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

In this paper we consider positive supersolutions of the nonlinear elliptic equation

$ - \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega, $

where $ 0\le p<1 $, $ \Omega $ is an arbitrary domain (bounded or unbounded) in $ {\mathbb{R}}^N $ ($ N\ge 2 $), $ f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+} $ $ (0 < a_{f} \leq +\infty) $ is a non-decreasing continuous function and $ \rho: \Omega \rightarrow \mathbb{R} $ 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 applications, 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 $. In particular when $ \rho(x) = |x|^\beta $ ($ \beta\in {\mathbb{R}} $) and $ f(u) = u^q $ with $ q+p>1 $ then every positive supersolution in an exterior domain is eventually constant if

$ (N-2)q+p(N-1)< N+\beta. $