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 $0\le p<1$, $\Omega$ is an arbitrary domain (bounded or unbounded) in $ \IR^N$ ($N\ge 2$), $f:[0,a_f)\to {\mathbb{R}}_{+}$ $(0<a_f⩽+\infty )$ 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 $x\in \Omega$ where $\nabla u\not\equiv 0$ 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 ${\sup }_{x\in \Omega }dist(x,\partial \Omega )=\infty$.