APDE seminar@BCAM: Existence of the extremals for the high order Hardy-Sobolev-Maz'ya inequalities

We establish the existence of the extremal function for the high order critical Hardy-Sobolev-Maz'ya (HSM) inequalities on the upper half space. The loss of  translation invariance and rearrangement technique on the upper half space,  together with the presence of the Hardy-singularity around the boundary of the upper half space for the HSM inequality cause much challenge to the study of the extremal problem. Instead of directly considering HSM inequality, we establish the existence of the extremal for its equivalent version: Poincare-Sobolev inequality on the hyperbolic space. We overcome the loss of compactness for minimizing sequence of Poincare-Sobolev inequality using Helgason-Fourier analysis, Riesz's rearrangement inequality and Lions concentration-compactness principle on the hyperbolic space. As an application, we also obtain the existence and symmetry of the positive solution for the high order Brezis-Nirenberg equation on the entire hyperbolic space.