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

We consider a nonlinear Schrödinger equation with double power nonlinearity

$i\partial _t u+\Delta u-\vert u\vert^{p-1}u+\vert u\vert^{q-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$

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 algebraic standing waves, as well as usual standing waves decaying exponentially with positive frequency. In this paper we study stability properties of two types of standing waves. We prove strong instability for all frequencies when $q\ge 1+4/N$ and instability for small frequencies when $q<1+4/N$, which especially give the first results on stability properties of algebraic standing waves. The instability result for small positive frequency when $q<1+4/N$ not only improves previous results in the one-dimensional case but also gives a first result on instability in the higher-dimensional case. The key point in our approach is to take advantage of algebraic standing waves.