  We consider a nonlinear Schr\"odinger 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 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 one-dimensional case but
also gives a first result on instability in higher-dimensional case. The key
point in our approach is to take advantage of algebraic standing waves.
