Asymptotic approach to singular solutions for the CR Yamabe equation

Abstract

Abstract We study the loss of compactness for optimal functions associated with subcritical approximations of the sharp Folland–Stein–Sobolev embedding in the Heisenberg group. A key ingredient is a fine asymptotic control of the optimal functions by the Jerison–Lee extremals, which attain equality in the critical Sobolev inequality ( JAMS , 1988). When boundary blow-up is ruled out, we show that centered-symmetric maximizers in a Korányi ball concentrate at its center, and we determine the exact blow-up rate and the pointwise asymptotic profile away from the pole, thereby obtaining a natural counterpart of the classical Atkinson–Brezis–Peletier asymptotics for nearly critical Sobolev problems. We then introduce a class of smooth domains geometrically regular near their characteristic set and extend the analysis to this setting, proving one-point concentration, describing the profile by the Dirichlet Green function, and expressing the exact blow-up rate through a Green-function boundary functional arising from the anisotropic Pohozaev identity.