Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

Abstract

In this survey paper, we consider variational problems involving the HardySchrödinger operator L γ := − − γ |x| 2 on a smooth domain of R n with 0 ∈ , and illustrate how the location of the singularity 0, be it in the interior of or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter can be stated as:where γ < n 2 4 , s ∈ [0, 2) and 2 (s) := 2(n−s) n−2 . We address questions regarding the explicit values of the optimal constant C := μ γ,s ( ), as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties often lead to situations where the best constants μ γ,s ( ) do not depend on the domain, and hence they are not attainable. We consider two different approaches for "breaking the homogeneity" of the problem, and restoring compactness. One approach was initiated by Brezis-Nirenberg, when γ = 0 and s = 0, and by Janelli, when γ > 0 and s = 0. It is suitable for the case where the singularity 0 is in the interior of , and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang for γ = 0, s > 0, and by C.S. Lin et al. and Ghoussoub-Robert, when γ = 0, s ≥ 0. It consists of considering domains, where the singularity 0 is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0 ∈ , then a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" theorem for the operator L γ is required in lower dimensions. If the singularity 0 belongs to the boundary ∂ , then the local geometry around 0 (convexity and mean curvature) plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" theorem is needed for the lower dimensions. Each case leads to a distinct notion of critical dimension for the operator L γ .

Sharp Hardy and Hardy–Sobolev inequalities with point singularities on the boundary

“…We note however that whereas in the interior singularity case the geometry of Ω is irrelevant, in this work the curvature of the boundary introduces several technical difficulties even in the case of the plain Hardy inequality (5) as already noted in several recent works see e.g. [8,9,10,11,14,15,20,21]. To overcome these difficulties we produce new improved inequalities in the flat case, see Lemmas 1, 2, 3 and then we use suitable conformal transformations thus obtaining sharp inequalities under the exterior ball assumption.…”

“…If on the other hand, the origin is on the boundary of a smooth near zero domain, then, related types of problems have been studied in [19,20,12,21]. More precisely the following minimization problem has been considered for 0 < s < 2 and n ≥ 4,…”

Existence of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity

“…The Hardy-Sobolev inequality is obtained by interpolating between the Sobolev inequality (s = 0) and the Hardy inequality (s = 2): For every s ∈ (0, 2) and n ≥ 3, there exists a positive constant K s,n such that [12], see also [13]. For s ∈ (0, 2), the best Hardy-Sobolev constant K s,n is attained by a one-parameter family (U η ) η>0 of functions where c n,s := ((n − s)(n − 2)) 1/(2 ⋆ (s)−2) is a positive normalising constant.…”

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

“…where Ω is a compact smooth subdomain of B n such that 0 ∈ Ω, but Ω does not touch the boundary of B n and λ ∈ R. Note that the metric is then smooth on such Ω, and the only singularity we will be dealing with will be coming from the Hardy-type potential V 2 and the Hardy-Sobolev weight V 2 * (s) , which behaves like 1 r 2 (resp., 1 r s ) at the origin. This is analogous to the Euclidean problem on bounded domains considered by Ghoussoub-Robert [12,11]. We shall therefore rely heavily on their work, at least in dimensions n ≥ 5.…”