On properties of positive solutions to nonlinear tri-harmonic and bi-harmonic equations with negative exponents

TL;DR

This paper explores the mathematical properties of positive solutions to certain nonlinear tri-harmonic and bi-harmonic equations with negative exponents, focusing on existence, behavior, and uniqueness, relevant to conformal geometry.

Contribution

It provides new insights into the properties of solutions to these complex equations, including nonexistence, asymptotics, and integral formulas, which were not previously fully understood.

Findings

Nonexistence results for certain parameter ranges

Asymptotic behavior characterizations

Integral representation formulas for solutions

Abstract

In this paper, we investigate various properties (e.g., nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear tri-harmonic equations in $\mathbb{R}^{n}$ ($n\geq2$) and bi-harmonic equations in $\mathbb{R}^{2}$ with negative exponents. Such kind of equations arise from conformal geometry.