Limit Behavior of Mass Critical Hartree Minimization Problems with Steep Potential Wells

TL;DR

This paper investigates the existence and behavior of minimizers in a mass critical Hartree energy problem with steep potential wells, revealing thresholds for minimizer existence and analyzing their concentration as potential strength increases.

Contribution

It establishes critical mass thresholds for minimizer existence and describes the asymptotic concentration behavior of minimizers as the potential becomes steep.

Findings

Existence of a critical mass N* beyond which no minimizers exist.

Existence of a λ*(N) threshold for minimizer existence depending on λ.

Minimizers concentrate at the potential well bottom as λ→∞.

Abstract

We consider minimizers of the following mass critical Hartree minimization problem: \[ e_\lambda(N):=\underset{\{u\in H^1(R^d),\,\|u\|^2_2=N\}}{\inf} E_\lambda(u),\,\ d\ge 3, \] where the Hartree energy functional $E_\lambda(u)$ is defined by \[ E_\lambda(u):=\int_{R ^d}|\nabla u(x)|^2dx+\lambda \int_{R ^d}g(x)u^2(x)dx-\frac{1}{2} \int_{R ^d}\int_{R ^d} \frac{u^2(x)u^2(y)}{|x-y|^2}dxdy,\,\ \lambda>0,\] and the steep potential $g(x)$ satisfies $0=g(0)=\inf _{R^d}g(x)\le g(x)\le 1$ and $1-g(x)\in L^{\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^*>0$, independent of $\lambda g(x)$, such that if $N\ge N^*$, then $e_\lambda(N)$ does not admit minimizers for any $\lambda >0$; if $0<N<N^*$, then there exists a constant $\lambda ^*(N)>0$ such that $e_\lambda(N)$ admits minimizers for any $\lambda >\lambda ^*(N)$, and $e_\lambda(N)$ does not admit minimizers for $0<\lambda <\lambda ^*(N)$. For any given $0<N<N^*$, the limit behavior of positive minimizers for $e_\lambda(N)$ is also studied as $\lambda\to\infty$, where the mass concentrates at the bottom of $g(x)$.