Positive solutions of quasilinear elliptic equations with Fuchsian potentials in Wolff class

TL;DR

This paper investigates positive solutions of quasilinear elliptic equations with Fuchsian-type singular potentials in the Wolff class, establishing Liouville-type theorems and analyzing asymptotic behaviors near singular points using Harnack's inequality and scaling techniques.

Contribution

It introduces new Liouville-type theorems and asymptotic analysis for positive solutions with Fuchsian potentials in the Wolff class, extending previous results to more general singularities.

Findings

Liouville-type theorems for solutions with Fuchsian potentials

Asymptotic behavior characterization near isolated singularities

Application of Harnack's inequality and scaling methods

Abstract

Using Harnack's inequality and a scaling argument we study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $\zeta \in \partial\Omega\cup\{\infty\}$ for the quasilinear elliptic equation $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{ in } \Omega,$$ where $\Omega$ is a domain in $\mathbb{R}^d$, $d\geq 2$, $1<p<d$, and $A=(a_{ij})\in L_{\rm loc}^{\infty}(\Omega; \mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. It is assumed that the potential $V$ belongs to a certain Wolff class and has a generalized Fuchsian-type singularity at an isolated point $\zeta\in \partial \Omega \cup \{\infty\}$.