摘要
We consider a free energy functional on Cartan–Hadamard manifolds and investigate the existence of its global minimizers. The energy functional consists of two components: an entropy (or internal energy) and an interaction energy modelled by an attractive potential. The two components have competing effects, as they favour spreading by linear diffusion and blow-up by non-local attractive interactions, respectively. We find necessary and sufficient conditions for ground states to exist, in terms of the behaviours of the attractive potential at infinity and at zero. In particular, for general Cartan–Hadamard manifolds, superlinear growth at infinity of the attractive potential prevents the spreading. The behaviour can be relaxed for homogeneous manifolds, for which only linear growth of the potential is sufficient for this purpose. As a key tool in our analysis, we develop a new logarithmic Hardy–Littlewood–Sobolev inequality on Cartan–Hadamard manifolds.