Journée de rentrée 2023 du GT CalVa

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

16 janv. 2023, 13:30

1h

Amphi Turing (Université Paris-Cité (campus Grands Moulins))

Orateur

Simona Rota Nodari (Laboratoire J.A. Dieudonné, Université Côte d’Azur)

Description

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 $-\mathrm{\Delta }u=g(u)$. 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. Joint work with Mathieu Lewin.