Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

Abstract

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- Δu = ρ(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } Ω,\] where $0\le p<1$, $ Ω$ 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 $ρ: Ω\rightarrow \IR$ is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions $u$ at each point $x\inΩ$ 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 $Ω$ with the property that $\sup_{x\inΩ}dist (x,\partialΩ)=\infty$.