A Perturbation Approach for Paneitz Energy on Standard Three Sphere

Abstract

We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetrization for higher order variational problems.

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

“…Recently, F. Hang and P. Yang [8] proposed another rearrangement method to prove the same inequality: If P S 3 is replaced by P S 3 + ǫ for some small constant ǫ > 0, then we study the minimizing problem:…”

“…Conjecture. (F. Hang and P. Yang [8]) If ǫ > 0 is a small constant and u ǫ is a positive smooth solution of P S 3 u ǫ + ǫu ǫ = −u −7 ǫ on S 3 , (1.5)…”