Concentration Behavior of Nonlinear Hartree-type Equation with almost Mass Critical Exponent

We study the following nonlinear Hartree-type equation \begin{equation*} -\Delta u+V(x)u-a(\frac{1}{|x|^\gamma}\ast |u|^2)u=\lambda u,~$\text{in}$~$\mathbb{R}$^N, \end{equation*} where $a>0$, $N\ge 3$, $\gamma \in (0,2)$ and $V(x)$ is an external potential. We first study the asymptotic behavior of the ground state of equation for $V(x)\equiv 1$, $a=1$ and $\lambda =0$ as $\gamma \nearrow 2$. Then we consider the case of some trapping potential $V(x)$, and show that all the mass of ground states concentrate at a global minimum point of $V(x)$ as $\gamma \nearrow 2$, which leads to symmetry breaking. Moreover, the concentration rate for maximum points of ground states will be given.