The Hardy-Schrödinger operator on the Poincaré Ball: compactness and multiplicity

Nassif Ghoussoub (1) , Saikat Mazumdar (1) , Frédéric Robert (2)

1 UBC Mathematics - Department of Mathematics [Vancouver]
2 IECL - Institut Élie Cartan de Lorraine

Let Ω B n be a a compact smooth domain in the Poincaré ball model of the Hyperbolic space B n , n ≥ 5. Let 0 < s < 2 and write 2 (s) := 2(n−s) n−2 for the corresponding critical Sobolev exponent. We show that if γ < (n−2) 2 4 − 4 and λ > n−2 n−4 n(n−4) 4 − γ , then the following Dirichlet boundary value problem: −∆ B n u − γV 2 u − λu = V 2 (s) |u| 2 (s)−2 u in Ω B n u = 0 on ∂Ω B n , has infinitely many solutions. Here −∆ B n is the Laplace-Beltrami operator associated with the metric g B n = 4 (1−|x| 2) 2 g Eucl , V 2 is the corresponding Hardy-type potential that behaves like 1 r 2 at the origin, while V 2 (s) is the Hardy-Sobolev weight, which behaves like 1 r s at the origin. The solutions belong to C 2 (Ω B n \ {0}) while around 0 they behave like u(x) ∼ K |x| n−2 2

Domaines: Equations aux dérivées partielles [math.AP]

Nassif Ghoussoub, Saikat Mazumdar, Frédéric Robert. The Hardy-Schrödinger operator on the Poincaré Ball: compactness and multiplicity. 2018. ⟨hal-01770803v1⟩