On a nonlinear Schrödinger equation: uniqueness, non-degeneracy and applications

Abstract: In this talk, I will first state a general result about the uniqueness and the non-degeneracy of positive radial solutions to some semi-linear elliptic equations $-\Delta u=g(u)$. Then I will consider the case of the double power non-linearity $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$. In this case, the non-degeneracy of the unique solution $u_\mu$ allows us to derive its behavior in the two limits $\mu \to 0$ and $\mu\to\mu_*$ where $\mu_*$ is the threshold of existence. This implies the uniqueness of energy minimizers at fixed mass in certain regimes. Moreover, for $\mu$ close to $\mu_*$, this gives some important information about the orbital stability of $u_\mu$ and allows us to provide a mathematical explanation for the occurence of a 'saturation phenomenon' which plays an important role in Physics.