具有局部排斥的奇摄动Choquard方程的极限轮廓
Limit profiles for singularly perturbed Choquard equations with local repulsion
摘要：
研究了形式为$$-Δu +\varepsilon u-(I_α*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$的Choquard型方程，其中$N\geq3$，$I_α$是Riesz势，具有$α\in(0,N)$，$p>1$，$q>2$和$\varepsilon\ge 0$ .这种类型的方程描述了自相互作用多体系统的集体行为。非局部非线性项表示长程吸引，而局部非线性项表示短程排斥。本文的第一部分在一个接近最优的参数范围内证明了$(P_0)$和$(P_\varepsilon)$与$\varepsilon>0$的正基态的存在性，并研究了正基态的正则性和定性性质。我们还研究了与$(P_\varepsilon)$相关的积分型Fermi方程的紧支撑基态的存在性。在本文的第二部分中，对于$\varepsilon\to 0$，我们确定了六个不同的渐近制度，并提供了一个表征的极限配置文件的基态$(P_\varepsilon)$在每个制度。我们还概述了三种不同的渐近制度的情况下$\varepsilon\to\infty$。在其中一个渐近区域中，$(P_\varepsilon)$的正基态收敛于一个紧支撑的P-Fermi极限轮廓。这是一种新的、纯粹的非定域现象，在具有$α=0$的$(P_\varepsilon)$的定域原型情况下无法观察到。特别是，这为自引力玻色-爱因斯坦凝聚体的天体物理模型中的费米子-费米近似提供了一个理由。
英文摘要：
We study Choquard type equation of the form $$-Δu +\varepsilon u-(I_α*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_α$ is the Riesz potential with $α\in(0,N)$, $p>1$, $q>2$ 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 $α=0$. In particular, this provides a justification for the Thomas-Fermi approximation in astrophysical models of self-gravitating Bose-Einstein condensate.
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