The lack of compactness usually happens for solutions of geometric PDEs which are invariant under certain groups. In this talk, we will consider conformally invariant problems. We will review the compactness and non-compactness results for the Yamabe equation. 

Second, we will report our recent compactness results for the constant Q-curvature equation, which is a fourth order analogue of the Yamabe equation.

Last, we will talk about our new ideas to overcome difficulties stemming from the fourth order equation. This is joint work with Professor YanYan Li.