Multipolar Hardy inequalities on Riemannian manifolds

TL;DR

This paper establishes multipolar Hardy inequalities on Riemannian manifolds, revealing how curvature influences these inequalities and applying them to Schrödinger problems to derive existence and non-existence results.

Contribution

It extends Euclidean multipolar Hardy inequalities to curved manifolds, incorporating curvature effects and applying variational and group-theoretic methods to Schrödinger equations.

Findings

Hardy inequalities depend on manifold curvature

Existence and non-existence results for Schrödinger problems

Application to Cartan-Hadamard and upper hemisphere manifolds

Abstract

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schr\"odinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.