We will investigate the effects of the lack of compactness of the critical Folland-Stein-Sobolev embedding by proving that a famous conjecture of Brezis and Peletier (Progr. Nonlinear Differential Equations Appl. 1989) still holds in the Heisenberg framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point which can be localized via the Green function associated to the involved domain. In order to achieve the aforementioned result we will combine several new estimates and specific tools to attack the related CR Yamabe equation (Jerison-Lee, J. Diff. Geom. 1987) with new feasible results in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as a De Giorgi's Gamma-convergence approach to provide fine energy approximations in very general (possible non-smooth) domains; Caccioppoli-type boundedness estimates depending on the datum for the solutions to even more general subelliptic equations; the asymptotic control of the optimal functions via the Jerison&Lee estremals realizing the equality in the critical Sobolev inequality (J. Amer. Math. Soc. 1988); the celebrated Global Compactness result which we will extend in the Heisenberg framework via a completely different approach with respect to the original one by Struwe (Math. Z. 1984). Il seminario si basa su un lavoro in collaborazione con Mirco Piccinini (Univ. Parma) e Letizia Temperini (Indam - Univ. Firenze).