Multipolar Hardy inequalities on Riemannian manifolds Dedicated to Professor Enrique Zuazua on the occasion of his 55th birthday

Faraci, Francesca; Farkas, Csaba; Kristály, Alexandru

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 551-567.

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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