Asymptotic approach to singular solutions for the CR Yamabe equation

TL;DR

This paper studies the concentration behavior of solutions to the CR Yamabe equation in the Heisenberg group, confirming a conjecture about energy concentration points and introducing new geometric domain regularity concepts.

Contribution

It proves a conjecture on energy concentration for the CR Yamabe equation in the Heisenberg setting and introduces a new domain regularity concept near characteristic sets.

Findings

Optimal functions concentrate energy at a single critical point.

The concentration point is a critical point of the Robin function.

New estimates and tools for the CR Yamabe equation in the Heisenberg framework.

Abstract

We investigate some effects of the lack of compactness in the critical Sobolev embedding by proving that a famous conjecture of Brezis and Peletier \cite{BP89} does still hold in the Heisenberg framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at exactly one point which is a critical point of the Robin function (i. e., the diagonal of the regular part of the Green function associated to the involved domain), in clear accordance with the underlying sub-Riemannian geometry. Consequently, a new suitable definition of domains geometrical regular near their characteristic set is introduced. In order to achieve the aforementioned result, we need to combine proper estimates and tools to attack the related CR Yamabe equation with novel feasible ingredients in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as for instance a fine asymptotic control of the optimal functions via the Jerison and Lee extremals realizing the equality in the critical Sobolev inequality \cite{JL88}.