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 \begin{document}$ i\partial_t\psi+\Delta\psi+\gamma|x|^{-\sigma}\psi+|\psi|^\alpha\psi = 0, \; \; \; (t, x)\in\mathbb{R}\times\mathbb{R}^N, $\end{document} where $ N\geq3 $, $ 0 < \gamma < \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 ] .