Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems
M. Á. Burgos-Pérez - Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna, Spain (email) 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 $\mathbb{R}^N$.
Keywords: Nonlinear elliptic equations, Liouville theorems, classification of solutions.
Received: May 2015; Revised: January 2016; Available Online: May 2016. |
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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
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