Threshold dynamics for the 3$d$ radial NLS with combined nonlinearity

TL;DR

This work classifies the dynamics of radial $H^1$ solutions to the 3D cubic-quintic NLS at the energy threshold $E(u)=E^c(W)$. By combining modulation around the ground state $W$, localized virial estimates, concentration-compactness, and a perturbative embedding of the quintic energy-critical model, the authors show that threshold solutions either scatter globally or blow up in finite time, with no heteroclinic thresholds in the $H^1$ setting. They further rule out nonscattering compact solutions by excluding both finite- and infinite-time blowup, thus extending Kenig–Merle-type threshold analysis to a mixed-nonlinearity NLS. The results illuminate ground-state stability under the cubic-quintic perturbation and provide a rigorous framework for threshold dynamics in focusing-defocusing NLS systems.

Abstract

We consider the nonlinear Schrödinger equation with focusing quintic and defocusing cubic nonlinearity in three space dimensions: \[ (i\partial_t+Δ)u = |u|^2 u - |u|^4 u. \] In [18, 23], the authors classified the dynamics of solutions under the energy constraint $E(u)< E^c(W)$, where $W$ is the quintic NLS ground state and $E^c$ is the quintic NLS energy. In this work we classify the dynamics of $H^1$ solutions at the threshold $E(u)=E^c(W)$.

Theorems & Definitions (63)

• Theorem 1.1: Sub-threshold dynamics, XuZhao2020MiaoXuZhao2013
• Theorem 1.2: Threshold dynamics
• Lemma 2.1: Strichartz estimates, GinibreVeloKeelTaoStrichartz
• Lemma 2.2: Bernstein inequalities
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6