Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

In this paper we consider positive supersolutions of the nonlinear elliptic equation $$-\mathrm{\Delta }u=\rho (x)f(u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p,\text{hfill}in\mathrm{\Omega },$$ where $0\le p<1$, $\mathrm{\Omega }$ is an arbitrary domain (bounded or unbounded) in ${\text{IR}}^N$ ($N\ge 2$), $f:[0,a_f)\to {\mathbb{R}}_{+}$ $(0 < a_{f} \leqslant +\infty)$isanon-decreasingcontinuousfunctionand$\rho: \Omega \rightarrow \IR$ is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions $u$ at each point $x\in \mathrm{\Omega }$ where $\mathrm{\nabla }u\equiv ̸0$ 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 $\mathrm{\Omega }$ with the property that ${\sup }_{x\in \mathrm{\Omega }}dist(x,\mathrm{\partial }\mathrm{\Omega })=\mathrm{\infty }$.