| Abstract: |
| We present some results regarding existence of non-trivial bound states of prescribed mass for the $L^2$-supercritical nonlinear Schr\odinger equation on metric graphs.
In recent years, the NLSE on graphs was studied by many authors in the $L^2$-subcritical or critical case. In parallel, the search for prescribed mass solutions to the $L^2$-supercritical NLSE in the Euclidean space attracted a lot of attention. However, the $L^2$-supercritical NLSE on graphs was essentially untouched. In such case, the mass constraint introduces severe complications in proving the existence of bounded Palais-Smale sequences. Several approaches have been developed to overcome these issues in the Euclidean case, but ultimately most of them seem to rely on the fact that critical points satisfy a natural constraint induced by a Pohozaev-type identity, on which the functional can be shown to be coercive. These methods allow to treat cases where the functional enjoys some nice scaling properties, but are not applicable if scaling is not allowed, such as on metric graphs.
In this talk we present some existence results obtained by developing a new method based upon a variational principle which combines the monotonicity trick and a min-max theorem with second order information for constrained functionals, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index. |
|