Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

TL;DR

This paper derives explicit pointwise estimates for positive supersolutions of a class of nonlinear elliptic equations involving gradient terms, with applications to dead core sets and Liouville theorems in unbounded domains.

Contribution

It provides new explicit estimates on supersolutions of nonlinear elliptic equations with gradient dependence, extending understanding of their behavior in various domain types.

Findings

Explicit estimates for supersolutions at points with non-zero gradient.

Characterization of dead core regions in bounded domains.

Liouville type results for unbounded domains with infinite inradius.

Abstract

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 $\sup_{x\in\Omega}dist (x,\partial\Omega)=\infty$.