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

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 $$P_{S^3}{u}_{\epsilon }+\epsilon u_{\epsilon }=-u_{\epsilon }^{-7}\mathrm{on}{S}^3,$$ 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) ].

22 pages

References in corpus (1)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)