On some conjectures proposed by Haı̈m Brezis Original Research Article

Druet (Ann. Inst. H. Poincaré Anal. Non Linèaire 19(2) (2002) 125) solved two conjectures proposed by Haı̈m Brezis (Comm. Pure Appl. Math. 39 (1986) 17) about “low”-dimension phenomena for some elliptic problem with critical Sobolev exponent. In Druet (Ann. Inst. H. Poincaré Anal. Non Linèaire 19(2) (2002) 125), the proof of one of the two conjectures is reduced to an asymptotic analysis which is carried over with very general techniques involving pointwise estimates. We propose here a different and simpler approach in the blow-up analysis based on integral estimates and on a careful expansion of the energy functional.