Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

M. Á.  Burgos-Pérez; J. García-Melián; A.  Quaas

doi:10.3934/dcds.2016004

Article Contents

2016, Volume 36, Issue 9: 4703-4721. Doi: 10.3934/dcds.2016004

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Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

1.
Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna
2.
Dpto. de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna
3.
Departamento de Matemática, Universidad Técnico Fedrico Santa María, Casilla V-110, Avda. España, 1680 - Valparaíso

Received: April 30, 2015

Revised: December 31, 2015

Published: May 2017

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Abstract

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 $\mathbb{R}^N$.

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Mathematics Subject Classification: Primary: 35J15, 35J60; Secondary: 35B53.

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