Limit profiles for singularly perturbed Choquard equations with local repulsion

TL;DR

This paper investigates the behavior of solutions to a nonlocal Choquard equation with local repulsion, analyzing existence, regularity, and asymptotic profiles of groundstates as the parameter varies, revealing new nonlocal phenomena.

Contribution

It introduces a detailed analysis of limit profiles for singularly perturbed Choquard equations, including novel nonlocal asymptotic regimes and their implications.

Findings

Existence and regularity of groundstates established.

Identification of six asymptotic regimes as e0 o 0.

Discovery of a nonlocal Thomas-Fermi limit profile.

Abstract

We study Choquard type equation of the form $$-\Delta u +\varepsilon u-(I_{\alpha}*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_\alpha$ is the Riesz potential with $\alpha\in(0,N)$, $p>1$, $q>2$ and $\varepsilon\ge 0$. 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 $(P_\varepsilon)$ with $\varepsilon>0$. We also study the existence of a compactly supported groundstate for an integral Thomas-Fermi type equation associated to $(P_\varepsilon)$. In the second part of the paper, for $\varepsilon\to 0$ we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of $(P_\varepsilon)$ in each of the regimes. We also outline three different asymptotic regimes in the case $\varepsilon\to\infty$. In one of the asymptotic regimes positive groundstates of $(P_\varepsilon)$ 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 $(P_\varepsilon)$ with $\alpha=0$. In particular, this provides a justification for the Thomas-Fermi approximation in astrophysical models of self-gravitating Bose-Einstein condensate.