Existence and dynamics of normalized solutions to Schr\"odinger equations with generic double-behaviour nonlinearities

We study the existence of solutions ( u,λᵤ)∈ H¹(RN; R) × R to \[ -Δ u + λ u = f(u) in R^N \] with N ≥ 3 and prescribed L² norm, and the dynamics of the solutions to \[ {cases} i ∂_t Ψ + Δ Ψ = f(Ψ)\\ Ψ(·,0) = ψ_0 ∈ H^1(R^N; C) {cases} \] with ψ₀ close to 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.