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We investigate some effects of the lack of compactness in the criticalSobolev embedding by proving that a famous conjecture of Brezis & Peletier(Essays in honor of Ennio De Giorgi -- Progr. Differ. Equ. Appl. 1989) doesstill hold in the Heisenberg framework: optimal functions for a naturalsubcritical approximations of the Sobolev quotient concentrate energy at onepoint which can be localized via the Green function associated to the involveddomain and in clear accordance with the underlying sub-Riemannian geometry --and consequently a new suitable definition of domains geometrical regular neartheir characteristic set is given. In order to achieve the aforementionedresult, we need to combine proper estimates and tools to attack the related CRYamabe equation (Jerison & Lee, J. Diff. Geom. 1987) with novel feasibleingredients in PDEs and Calculus of Variations which also aim to constitutegeneral independent results in the Heisenberg framework, as e.g. a fineasymptotic control of the optimal functions via the Jerison & Lee extremalsrealizing the equality in the critical Sobolev inequality (J. Amer. Math. Soc.1988).

Asymptotic approach to singular solutions for the CR Yamabe equation, and a conjecture by H. Brezis and L. A. Peletier in the Heisenberg group