Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

Autor: Jorge García-Melián, M. Á. Burgos-Pérez, Alexander Quaas
Rok vydání: 2016
Předmět:
Zdroj: Discrete and Continuous Dynamical Systems. 36:4703-4721
ISSN: 1078-0947
DOI: 10.3934/dcds.2016004
Popis: In this paper we consider positive supersolutions of the elliptic equation $-\triangle u = f(u) |\nabla u|^q$, posed in exterior domains of $\mathbb{R}^N$ ($N\ge 2$), where $f$ is continuous in $[0,+\infty)$ and positive in $(0,+\infty)$ and $q>0$. We classify supersolutions $u$ into four types depending on the function $m(R)=\inf_{|x|=R} u(x)$ for large $R$, and give necessary and sufficient conditions in order to have supersolutions of each of these types. As a consequence, we also obtain Liouville theorems for supersolutions depending on the values of $N$, $q$ and on some integrability properties on $f$ at zero or infinity. We also describe these questions when the equation is posed in the whole $\mathbb{R}^N$.
Databáze: OpenAIRE