Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities

Abstract

We study the existence of solutions $(\underline{u},{\lambda }_{\underline{u}})\in H^1({\mathbb{R}}^N;\mathbb{R})\times \mathbb{R}$ to \[ -Δu + λu = f(u) \quad \text{in } \mathbb{R}^N \] with $N\ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t Ψ+ ΔΨ= f(Ψ)\\ Ψ(\cdot,0) = ψ_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] with ${\psi }_0$ close to $\underline{u}$. Here, the nonlinear term $f$ has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution.

27 pages, minor corrections