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We consider a double power nonlinear Schr\"odinger equation which possessesthe algebraically decaying stationary solution $\phi_0$ as well asexponentially decaying standing waves $e^{i\omega t}\phi_\omega(x)$ with$\omega>0$. It is well-known from the general theory that stability propertiesof standing waves are determined by the derivative of $\omega\mapstoM(\omega):=\frac{1}{2}\|\phi_\omega\|_{L^2}^2$; namely $e^{i\omegat}\phi_\omega$ with $\omega>0$ is stable if $M'(\omega)>0$ and unstable if$M'(\omega)<0$. However, the stability/instability of stationary solutions isoutside the general theory from the viewpoint of spectral properties oflinearized operators. In this paper we prove the instability of the stationarysolution $\phi_0$ in one dimension under the condition$M'(0):=\lim_{\omega\downarrow 0}M'(\omega)\in[-\infty, 0)$. The key in theproof is the construction of the one-sided derivative of$\omega\mapsto\phi_\omega$ at $\omega=0$, which is effectively used toconstruct the unstable direction of $\phi_0$.

Instability of stationary solutions for double power nonlinear Schr\"odinger equations in one dimension