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Existence of ground states for free energies on the hyperbolic space
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 07 August 2025, pp. 1-43
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We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space
${\mathbb H}^n$. 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 
${\mathbb H}^n$. 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.
