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

TL;DR

This paper establishes Liouville-type theorems and describes the asymptotic behavior of positive solutions near isolated singularities for a class of quasilinear elliptic equations with Fuchsian potentials in Morrey spaces.

Contribution

It introduces new Liouville theorems and asymptotic analysis for solutions of $(p,A)$-Laplacian equations with Fuchsian potentials, extending previous results to Morrey space potentials.

Findings

Positive solutions exhibit specific asymptotic behavior near singularities.

Liouville-type theorems restrict the form of solutions in certain domains.

The potential's Fuchsian singularity influences solution behavior significantly.

Abstract

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 $\zeta$.