Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems
Autor: | Jorge García-Melián, M. Á. Burgos-Pérez, Alexander Quaas |
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Rok vydání: | 2016 |
Předmět: |
Pure mathematics
Applied Mathematics media_common.quotation_subject 010102 general mathematics Mathematics::Analysis of PDEs Zero (complex analysis) Order (ring theory) Function (mathematics) Infinity 01 natural sciences 010101 applied mathematics Elliptic curve Nonlinear system Discrete Mathematics and Combinatorics Nabla symbol 0101 mathematics Analysis media_common Mathematics |
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 |
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