An action approach to nodal and least energy normalized solutions for nonlinear Schrödinger equations

by Damien Galant

Wednesday Jun 25, 2025, 2:00 PM → 3:00 PM Europe/Paris

Salle Huron (1R1-106)

In this talk, I will present two notions of stationary state solutions to the nonlinear Schrödinger equation:

those with a fixed frequency, corresponding to critical points of the action functional,

and those with fixed mass (normalized solutions), corresponding to critical points

of the energy functional constrained on a L²-sphere.

In general, it is somewhat easier to treat the problem with a fixed frequency.

In particular, in this case one is able to find least action solutions

and least action nodal solutions for all Sobolev-subcritical exponents in the nonlinearity.

Regarding the fixed mass solutions, a new critical exponent appears.

Finding normalized solutions in the "mass-supercritical" regime is usually a difficult problem,

explored since pioneering work by Jeanjean in the late 1990s and often imposing geometrical conditions

on the domain on which the equation is set. We will present a new method which allows

to characterize the masses of the least action solutions and the least action normalized solutions,

therefore building a bridge between the study of solutions having a fixed frequency and those having a fixed mass.

As we will see, we will do so in a new "variational" fashion since one does not expect in general

to have continuous branches of solutions for all values of the frequency.

This is joint work with Colette De Coster (CERAMATHS/DMATHS, UPHF and INSA HdF, Valenciennes, France),

Simone Dovetta (Politecnico di Torino, Italy) and Enrico Serra (Politecnico di Torino).