Uniqueness and nondegeneracy of ground states for nonlinear Schr\"odinger equations with attractive inverse-power potential

We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schr"odinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.