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

TL;DR

This paper investigates the existence, stability, and dynamics of solutions to nonlinear Schrödinger equations with complex nonlinearities that exhibit different growth behaviors at zero and infinity.

Contribution

It introduces new results on the existence, stability, and instability of normalized solutions for Schrödinger equations with general double-behavior nonlinearities.

Findings

Existence of a locally least-energy solution.

Orbital stability of these solutions.

Strong instability of certain solutions.

Abstract

We study the existence of solutions $(\underline u,\lambda_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and the dynamics of the solutions to \[ \begin{cases} \mathrm{i} \partial_t \Psi + \Delta \Psi = f(\Psi)\\ \Psi(\cdot,0) = \psi_0 \in H^1(\mathbb{R}^N; \mathbb{C}) \end{cases} \] 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.