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

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