Positive Liouville theorems and asymptotic behavior for p-Laplacian type elliptic equations with a Fuchsian potential

TL;DR

This paper investigates positive solutions of p-Laplacian elliptic equations with Fuchsian-type singular potentials, establishing Liouville theorems and analyzing their asymptotic behaviors near singularities or at infinity.

Contribution

It introduces new Liouville theorems and asymptotic analysis for p-Laplacian equations with Fuchsian singular potentials, extending classical results to singular and boundary cases.

Findings

Established conditions for positivity of solutions.

Derived asymptotic behaviors near singularities and at infinity.

Extended Liouville theorems to Fuchsian potential cases.

Abstract

We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u):= - pLaplace(u) + V |u|^{p-2} u = 0 in X, where X is a domain in R^d, d > 1, and 1<p<infty. We assume that the potential V has a Fuchsian type singularity at a point zeta, where either zeta=infty and X is a truncated C^2-cone, or zeta=0 and zeta is either an isolated point of a boundary of X or belongs to a C^2-portion of the boundary of X.