QLunch: Simona Rota Nodari

Speaker: Simona Rota Nodari, Université Côte d'Azur

Title: 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 certain regimes of the parameter $\mu$. This implies the uniqueness of energy minimizers at fixed mass and gives some important information about the orbital stability of $u_\mu$.

Joint work with Mathieu Lewin.