Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

Abstract

We consider a nonlinear Schrödinger equation with double power nonlinearity \begin{align*} i\partial _t u+\Delta u-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in \mathbb {R}\times \mathbb {R}^N, \end{align*} where $1<p<q<1+4/(N-2)_+$. Due to the defocusing effect from the lower power order nonlinearity, the equation has algebraically decaying standing waves with zero frequency, which we call

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