On the standing wave for a class of nonlinear Schrödinger equations

Abstract

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations i φ t + Δ φ − | x | 2 φ + μ | φ | p − 1 φ + γ | φ | q − 1 φ = 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.