Instability of Standing Waves to the Inhomogeneous Nonlinear Schr\"odinger Equation with Harmonic Potential

We study the instability of standing-wave solutions $e^{i\omega t}\phi_{\omega}(x)$totheinhomogeneousnonlinearSchr\ddot{o}dingerequation$$i\phi_t=-\triangle\phi+|x|^2\phi-|x|^b|\phi|^{p-1}\phi, \qquad \in$\mathbb{R}$^N, $$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 $b>0.$