  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 $g(x)$.

Functional Analysis;Analysis of PDEs