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In this paper we first prove a general result about the uniqueness andnon-degeneracy of positive radial solutions to equations of the form $\Deltau+g(u)=0$. Our result applies in particular to the double power non-linearitywhere $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$, which we discuss with moredetails. In this case, the non-degeneracy of the unique solution $u_\mu$ allowsus to derive its behavior in the two limits $\mu\to0$ and $\mu\to\mu_*$ where$\mu_*$ is the threshold of existence. This gives the uniqueness of energyminimizers at fixed mass in certain regimes. We also make a conjecture aboutthe variations of the $L^2$ mass of $u_\mu$ in terms of $\mu$, which weillustrate with numerical simulations. If valid, this conjecture would implythe uniqueness of energy minimizers in all cases and also give some importantinformation about the orbital stability of $u_\mu$.

The double-power nonlinear Schr\"odinger equation and its generalizations: uniqueness, non-degeneracy and applications