On the asymptotic behavior of radial entire solutions for the equation $(-\Delta)^3 u=u^p$ in $\mathbf{R}^n$

TL;DR

This paper investigates the existence and precise growth rates at infinity of radial positive solutions to a sixth-order polyharmonic equation in high-dimensional space, extending previous work with new methods.

Contribution

It introduces a sub- and super-solution method combined with comparison principles to classify the asymptotic behavior of solutions for the sixth-order equation.

Findings

Established existence of solutions for $p$ above the Joseph-Lundgren exponent.

Classified the exact growth rates of solutions at infinity.

Extended classification results to higher-order polyharmonic equations.

Abstract

Our main task in this note is to prove the existence and to classify the exact growth at infinity of radial positive $C^6$-solutions of $(-\Delta )^3 u = u^p$ in $\mathbf{R}^n$, where $n\geqslant 15$ and $p$ is bounded from below by the sixth-order Joseph-Lundgren exponent. Following the main work of Winkler, we introduce the sub- and super-solution method and comparision principle to conclude the asymptotic behavior of solutions.