Titre de la dissertation: « The nonlinear Schrödinger equation on metric graphs »

Résumé de la dissertation : In this thesis, we investigate the nonlinear Schrödinger equation

−Δu + λu = |u|p−2u (NLS)

where Δ :=P1≤i≤N ∂ii is the Laplacian, p, λ ∈ R and p > 2. The equation

will be set on open domains of RN or, in most chapters, on metric graphs.

To begin with, we set the stage in which the following chapters take place.

Thus, we present the superlinear elliptic equation (NLS), metric graphs and

the formulation of (NLS) on them.

Then, we introduce several notions. In particular, we consider two ways

to tackle the problem variationally: one based on the critical points of the

action functional on the Nehari manifold, leading to (nodal) action ground

states, the other based on critical points of the energy functional on a L2-mass

constraint, leading to normalized solutions. Five chapters follow, dedicated

to:

1. an existence theorem of solutions to (NLS) on metric graphs which

allows to construct examples where one may compare the notions of

action ground state and of minimal action solution on noncompact

domains;

2. existence and non-existence results for action ground states and nodal

action ground states on several classes of metric graphs;

3. a new method to prove the existence of (positive and nodal) L2-normalized

solutions to (NLS) with the Dirichlet boundary condition on

bounded open sets of RN, including in the L2-supercritical regime;

4. the infinite multiplicity of normalized solutions, on metric graphs and

in the L2-supercritical regime, to the nonlinear Schr¨odinger equation

with localized nonlinearity;

5. the asymptotic analysis of (NLS) on compact graphs in the asymptotic

regime p → 2 thanks to a Lyapunov-Schmidt reduction, the study of

nodal ground states vanishing identically on edges on compact star

graphs as well as the detailed study of the “tetrahedron graph” thanks

to a computer-assisted proof using computations certified by interval

arithmetic.