We consider a double power nonlinear Schrödinger equation possessing the algebraically decaying stationary solution \(\phi _0\) as well as exponentially decaying standing waves \(e^{i\omega t}\phi _\omega (x)\) with \(\omega >0\) . According to the general theory, stability properties of standing waves are determined by the derivative of \(\omega \mapsto M(\omega )\mathrel {\mathop :}=\frac{1}{2}\Vert \phi _\omega \Vert _{L^2}^2\) ; namely \(e^{i\omega t}\phi _\omega \) with \(\omega >0\) is stable if \(M'(\omega )>0\) and unstable if \(M'(\omega )<0\) . However, the stability/instability of stationary solutions is outside the general theory from the viewpoint of spectral properties of linearized operators. In this paper we prove the instability of the stationary solution \(\phi _0\) in one dimension under the condition \(\lim _{\omega \downarrow 0}M'(\omega )\in [-\infty , 0)\) . The key in the proof is the construction of the one-sided derivative of \(\omega \mapsto \phi _\omega \) at \(\omega =0\) , which is effectively used to construct the unstable direction.
