Limit profiles for singularly perturbed Choquard equations with local repulsion.

We study Choquard type equation of the form where N ≥ 3 , I α is the Riesz potential with α ∈ (0 , N) , p > 1 , q > 2 and ε ≥ 0 . Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of (P 0) and of (P ε) with ε > 0 . We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to (P ε) . In the second part of the paper, for ε → 0 we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of (P ε) in each of the regimes. We also outline three different asymptotic regimes in the case ε → ∞ . In one of the asymptotic regimes positive groundstates of (P ε) converge to a compactly supported Thomas–Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of (P ε) with α = 0 . In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate. [ABSTRACT FROM AUTHOR]