On the standing wave for a class of nonlinear Schrödinger equations

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equationsiφt+Δφ−|x|2φ+μ|φ|p−1φ+γ|φ|q−1φ=0,iφt+Δφ−|x|2φ+μ|φ|p−1φ+γ|φ|q−1φ=0, which describes the attractive Bose–Einstein condensates under a magnetic trap. We establish the existence of the standing wave of the equation. Furthermore, we prove that the standing wave is nonlinearly unstable.