We study the instability of standing-wave solutions $e^{ i\omega t}\phi_{ \omega}(x)$ to the inhomogeneous nonlinear Schr\"{ o}dinger equation $$i\phi_t=-\triangle\phi+|x|^2\phi-|x|^b|\phi|^{ p-1}\phi, \qquad \in\mathbb{ R}^N, $$ where $ b > 0 $ and $ \phi_{ \omega} $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $\omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
Categories: math.AP