Limit profiles for singularly perturbed Choquard equations with local repulsion

Abstract

We study Choquard type equation of the form $$-\Delta u+\epsilon u-(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u+|u{|}^{q-2}u=0in{\mathbb{R}}^N,(P_{\epsilon })$$ where $N\ge 3$, $I_{\alpha }$ is the Riesz potential with $\alpha \in (0,N)$, $p>1$, $q>2$ and $\epsilon \ge 0$. 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 $(P_{\epsilon })$ 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 $\epsilon \to 0$ 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 $\epsilon \to \infty$. 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.

arXiv comment47 pages; corrected typos and updated references

References in corpus (11)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• Can dark matter be a Bose-Einstein condensate?
• Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results
• A guide to the Choquard equation
• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
• Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency
• On fractional Schrodinger systems of Choquard type
• The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications
• Asymptotic properties of ground states of scalar field equations with a vanishing parameter
• Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsaecker model
• A New Estimate on the Two-Dimensional Indirect Coulomb Energy