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We study the stability/instability of standing waves for the one dimensionalnonlinear Schr\"odinger equation with double power nonlinearities:\begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad(t,x)\in \mathbb{R}\times\mathbb{R} ,~1<p<q. \end{align*} When $q<5$, thestability properties of standing waves $e^{i\omega t}\phi_{\omega}$ may changefor the frequency $\omega$. A sufficient condition for yielding instability forsmall frequencies are obtained in previous results, but it has not been knownwhat the sharp condition is. In this paper we completely calculate the explicitformula of $\lim_{\omega\to0}\partial_{\omega}\|\phi_{\omega}\|_{L^2}^2$, whichis independent of interest, and establish the sharp thresholds for stabilityand instability of standing waves.

Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schr\"odinger equation