Mass and Extremals Associated with the Hardy–Schrödinger Operator on Hyperbolic Space

Abstract

We consider the Hardy-Schrödinger operator Lγ := −∆ B n − γV 2 on the Poincaré ball model of the Hyperbolic space B n (n ≥ 3). Here V 2 is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V 2 (r) ∼ 1 r 2 . Just like in the Euclidean setting, the operator Lγ is positive definite whenever γ < (n−2) 2 4 , in which case we exhibit explicit solutions for the Sobolev critical equation Lγ u = V 2 * (s) u 2 * (s)−1 in B n , where 0 ≤ s < 2, 2 * (s) = 2(n−s) n−2 , and V 2 * (s) is a weight that behaves like 1 r s around 0. In dimensions n ≥ 5, the above equation in a domain Ω of B n containing 0 and away from the boundary, has a ground state solution, whenever 0 < γ ≤ n(n−4) 4 , and provided Lγ is replaced by a linear perturbation Lγ − λu, where λ > n − 2 n − 4 n(n − 4) 4 − γ . On the other hand, in dimensions 3and 4, the existence of solutions depends on whether the domain has a postive "hyperbolic mass," a notion that we introduce and analyze therein.

Hardy's Identities and Inequalities on Cartan-Hadamard Manifolds

“…It is also worth mentioning that the problems of improving Hardy type inequalities as well as other functional and geometric inequalities using the effect of curvature have been studied intensively recently. We refer the interested reader to [8,16,20,27,28,38,39,43], to name just a few.…”

Best Sobolev constants in the presence of sharp Hardy terms in Euclidean and hyperbolic space

“…One geometric environment where there has been a lot of recent activity on Hardy and Sobolev inequalities is the hyperbolic space H n ; see [3,7,8,9,12,13,14,16,17,18,19]. The analogue of the Sobolev inequality (2) in the hyperbolic space reads…”

“…where 2 < p ≤ 2 * and S n,p denotes the best constant. The positivity of S n,p follows from the positivity of S n,2 * (see [17]) together with (9).…”