Limit profiles for singularly perturbed Choquard equations with local repulsion

We study Choquard type equation of the form -Delta u + epsilon u - (I-alpha (*) vertical bar u vertical bar(p))vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(q-2)u = 0 in R-N, where N >= 3, I-alpha is the Riesz potential with alpha is an element of (0, N), p > 1, q > 2 and epsilon >= 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-epsilon) with epsilon > 0. We also study the existence of a compactly supported groundstate for an integral Thomas-Fermi type equation associated to (P-epsilon). In the second part of the paper, for epsilon -> 0 we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of (P-epsilon) in each of the regimes. We also outline three different asymptotic regimes in the case epsilon -> infinity. In one of the asymptotic regimes positive groundstates of (P-epsilon) 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-epsilon) with alpha = 0. In particular, this provides a justification for the Thomas-Fermi approximation in astrophysical models of self-gravitating Bose-Einstein condensate.