  In this paper we first prove a general result about the uniqueness and
non-degeneracy of positive radial solutions to equations of the form $\Delta
u+g(u)=0$. Our result applies in particular to the double power non-linearity
where $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$, which we discuss with more
details. In this case, the non-degeneracy of the unique solution $u_\mu$ allows
us 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 energy
minimizers at fixed mass in certain regimes. We also make a conjecture about
the variations of the $L^2$ mass of $u_\mu$ in terms of $\mu$, which we
illustrate with numerical simulations. If valid, this conjecture would imply
the uniqueness of energy minimizers in all cases and also give some important
information about the orbital stability of $u_\mu$.
