Compactness and non-compactness theorems of the fourth- and sixth-order constant Q-curvature problems

非线性泛函分析与偏微分方程学术报告

We will discuss some results about compactness and non-compactness in conformal geometry. Firstly, we prove that the solution set of the fourth-order constant $Q$-curvature problem is $C^4$-compact in dimensions $5 \le n \le 24$. For $n \ge 25$, an example of an $L^{\infty}$-unbounded sequence of solutions has been known for over a decade (Wei and Zhao). Secondly, we demonstrate that the solution set of the sixth-order constant $Q$-curvature problem is $C^6$-compact in dimensions $7 \le n \le 26$, whereas a blow-up example exists for $n \ge 27$.