Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation

In this paper, we consider the instability of standing waves for an inhomogeneous Gross-Pitaevskii equation $ i\psi_t +\Delta \psi -a^2|x|^2\psi +|x|^{-b}|\psi|^{p}\psi = 0. $ This equation arises in the description of nonlinear waves such as propagation of a laser beam in the optical fiber. We firstly proved that there exists $\omega_* \gt 0$ such that for all $\omega \gt \omega_*$, the standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is unstable. Then, we deduce that if $\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda = 1}\leq 0$, the ground state standing wave $e^{i\omega t}u_\omega(x)$ is strongly unstable by blow-up, where $u_\omega^\lambda(x) = \lambda^{\frac{N}{2}}u_\omega(\lambda x)$ and $S_\omega$ is the action. This result is a complement to the partial result of Ardila and Dinh (Z. Angew. Math. Phys. 2020), where the strong instability of standing waves has been studied under a different assumption.