Instability of Standing Waves to the Inhomogeneous Nonlinear Schrödinger Equation with Harmonic Potential

We study the instability of standing-wave solutions $e^{iωt}ϕ_ω(x)$ to the inhomogeneous nonlinear Schrödinger equation $iϕ_t=-\triangleϕ+|x|^2ϕ-|x|^b|ϕ|^{p-1}ϕ, \qquad \in\mathbb{R}^N,$ where $b > 0$ and $ϕ_ω$ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $ω$ of wave and the power of nonlinearity $p$ for any fixed $b > 0.$

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