Limit profiles for singularly perturbed Choquard equations with local repulsion

Abstract

AbstractWe study Choquard type equation of the form where $$N\ge 3$$ N ≥ 3 , $$I_\alpha $$ I α is the Riesz potential with $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) , $$p>1$$ p > 1 , $$q>2$$ q > 2 and $$\varepsilon \ge 0$$ 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 $$(P_0)$$ ( P 0 ) and of $$(P_\varepsilon )$$ ( P ε ) with $$\varepsilon >0$$ ε > 0 . We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to $$(P_{\varepsilon })$$ ( P ε ) . In the second part of the paper, for $$\varepsilon \rightarrow 0$$ ε → 0 we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of $$(P_\varepsilon )$$ ( P ε ) in each of the regimes. We also outline three different asymptotic regimes in the case $$\varepsilon \rightarrow \infty $$ ε → ∞ . In one of the asymptotic regimes positive groundstates of $$(P_\varepsilon )$$ ( 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 $$(P_\varepsilon )$$ ( P ε ) with $$\alpha =0$$ α = 0 . In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate.