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

Abstract

We study the existence of solutions $(\underset{\_}{u},{\lambda }_{\underset{\_}{u}})\in H^1(R^N;R)\times R$ to$-\Delta u+\lambda u=f\left(u\right)\text{in }{R}^N$ with $N\ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to$\{\begin{array}{c}i{\partial }_t\Psi +\Delta \Psi =f\left(\Psi \right)\\ \Psi (\cdot ,0)={\psi }_0\in H^1(R^N;C)\end{array}$ with ${\psi }_0$ close to $\underset{\_}{u}$. 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.