Concentration behavior of nonlinear Hartree-type equation with almost mass critical exponent

Abstract

We study the following nonlinear Hartree-type equation $$\begin{aligned} -\Delta u+V(x)u-a\left( \frac{1}{|x|^\gamma }*|u|^2\right) u=\lambda u,~\text {in}~{\mathbb {R}}^N, \end{aligned}$$ where $$a>0$$ , $$N\ge 3$$ , $$\gamma \in (1,2)$$ 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.