Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials

In this paper, we consider the following nonlinear Schrödinger equation with attractive inverse-power potentials

$$i{\mathrm{\partial }}_t\psi +\mathrm{\Delta }\psi +\gamma |x{|}^{-\sigma }\psi +|\psi {|}^{\alpha }\psi =0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$

where $N\ge 3$, $0<\gamma <\mathrm{\infty }$, $0<\sigma <2$ and $\frac{4}{N}<\alpha <\frac{4}{N-2}$. By using the concentration compactness principle and considering a local minimization problem, we prove that there exists a ${\gamma }_0>0$ sufficiently small such that $0<\gamma <{\gamma }_0$ and for any $a\in (0,a_0)$, there exist stable standing waves for the problem in the $L^2$-supercritical case. Our results are complement to the result of Li-Zhao in ^[23].