Energy asymptotics in the Brezis–Nirenberg problem: The higher-dimensional case
Résumé
For dimensions N ě 4, we consider the Brézis-Nirenberg variational problem of finding Sp V q :" inf 0ıuPH 1 0 pΩq ş Ω |∇u| 2 dx` ş Ω V |u| 2 dx ş Ω |u| q dx¯2 {q , where q " 2N N´2 is the critical Sobolev exponent, Ω Ă R N is a bounded open set and V : Ω Ñ R is a continuous function. We compute the asymptotics of Sp0q´Sp V q to leading order as Ñ 0`. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case N " 3.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)