Existence of ground states for free energies on the hyperbolic space

TL;DR

This paper studies the existence of ground states for a free energy functional involving nonlocal interactions and local repulsion on hyperbolic space, deriving conditions and inequalities relevant to aggregation phenomena.

Contribution

It establishes necessary and sufficient conditions for ground states to exist on hyperbolic space and derives Hardy-Littlewood-Sobolev inequalities on Cartan-Hadamard manifolds.

Findings

Conditions for ground state existence on hyperbolic space

Derivation of HLS-type inequalities on Cartan-Hadamard manifolds

Insights into aggregation-diffusion models in curved spaces

Abstract

We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space $\bbh^\dm$. The free energy consists of two competing terms: an entropy, corresponding to slow nonlinear diffusion, that favours spreading, and an attractive interaction potential energy that favours aggregation. We establish necessary and sufficient conditions on the interaction potential for ground states to exist on the hyperbolic space $\bbh^\dm$. To prove our results we derived several Hardy-Littlewood-Sobolev (HLS)-type inequalities on general Cartan-Hadamard manifolds of bounded curvature, which have an interest in their own.