Positive Liouville theorem and asymptotic behaviour for \((p,A)\)-Laplacian type elliptic equations with Fuchsian potentials in Morrey space

We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point \(\zeta\in\partial\Omega\cup\{\infty\}\) of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } \Omega\setminus\{\zeta\},$$ wher...

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Bibliographic Details
Published in:arXiv.org 2020-07
Main Authors: Ratan Kr Giri, Pinchover, Yehuda
Format: Article
Language:English
Subjects:
Asymptotic properties
Elliptic functions
Liouville theorem
Mathematical analysis
Matrix methods
Singularity (mathematics)
Online Access:Get full text
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Summary:We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point \(\zeta\in\partial\Omega\cup\{\infty\}\) of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } \Omega\setminus\{\zeta\},$$ where \(\Omega\) is a domain in \(\mathbb{R}^d\) (\(d\geq 2\)), and \(A=(a_{ij})\in L_{\rm loc}^{\infty}(\Omega;\mathbb{R}^{d\times d})\) is a symmetric and locally uniformly positive definite matrix. The potential \(V\) lies in a certain local Morrey space (depending on \(p\)) and has a Fuchsian-type isolated singularity at \(\zeta\).
ISSN:2331-8422