Multipolar Hardy inequalities on Riemannian manifolds

2016/09/05 by Francesca Faraci, Faraci, Francesca, Csaba Farkas +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1609.01080

18 pages; accepted for publication in ESAIM - Control, Optimisation and Calculus of Variations

arxiv created 2017/08/27 · arxiv updated 2017/08/29

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 Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

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