A Liouville type theorem of the linearly perturbed Paneitz equation on $S^3$

TL;DR

This paper proves a Liouville type theorem for a perturbed Paneitz equation on the 3-sphere, confirming a conjecture and showing that positive solutions must be constant when the perturbation parameter is small.

Contribution

It establishes a Liouville theorem for the linearly perturbed Paneitz equation on $S^3$, confirming a conjecture by Hang and Yang.

Findings

Positive solutions are constant for small perturbations.

Confirms a conjecture by Hang and Yang.

Advances understanding of Paneitz equations on spheres.

Abstract

We prove a Liouville type theorem for the linearly perturbed Paneitz equation: For $\epsilon>0$ small enough, if $u_\epsilon$ is a positive smooth solution of $$P_{S^3} u_\epsilon+\epsilon u_\epsilon=-u_\epsilon^{-7} \qquad \mathrm{~~on~~}S^3,$$ where $P_{S^3}$ is the Paneitz operator of the round metric $g_{S^3}$, then $u_\epsilon$ is constant. This confirms a conjecture proposed by Fengbo Hang and Paul Yang in [ Int. Math. Res. Not. IMRN, 2020 (11) ].