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

We study the instability of standing-wave solutions e iωt φ ω (x) to the inhomogeneous nonlinear Schrödinger equation iϕ t = −△ϕ + |x| 2 ϕ − |x| b |ϕ| p−1 ϕ, x ∈ 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.