Radial 3D Focusing Energy Critical INLS equations with defocusing perturbation: Ground states, Scattering, and Blow-up

TL;DR

This work analyzes the radial Cauchy problem for the focusing energy-critical INLS with a defocusing perturbation in three dimensions: $i\partial_t u+\Delta u=|x|^{-a}|u|^{p-2}u-|x|^{-b}|u|^{6-2b}u$ with $0<a,b<2$ and $2+\frac{4-2a}{3}<p<6-2a$. It develops a variational framework around a Pohozaev manifold to characterize ground states, proves existence/nonexistence and quantitative properties of ground states, and establishes a dichotomy between scattering and blow-up below the ground-state energy. The analysis combines energy/mass conservation, radial Strichartz theory, Virial/Morawetz estimates, and Dodson–Murphy-style scattering arguments adapted to the inhomogeneous, non-translation-invariant setting. Under radial symmetry and specific parameter ranges, the authors prove global well-posedness and scattering for data in the ground-state sublevel and show finite-time blow-up for data with negative Virial/energy below threshold. Overall, the paper advances understanding of dynamics under competing inhomogeneous nonlinearities and provides a framework applicable to related INLS models in nonlinear optics, plasma physics, and Bose–Einstein condensates.

Abstract

We investigate the following inhomogeneous nonlinear Schrödinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +Δu =|x|^{-a} |u|^{p-2} u - |x|^{-b} |u|^{4-2b}u \quad \mbox{in} \,\, \mathbb{R}_t \times \mathbb{R}_x^3, $$ where $0<a$, $b<2$ and $2+\frac{4-2a}{3}< p\leq 6-2a$. First, we establish the existence and nonexistence of ground states, along with their quantitative properties. Subsequently, we analyze the dichotomy between scattering and blow-up for solutions with energy below the ground-state energy threshold. An intriguing feature of this equation is the lack of scaling invariance, which arises from the competing effects of the inhomogeneous nonlinearities. Additionally, the presence of singular weights breaks translation invariance in the spatial variable, introducing further complexity to the analysis. To the best of our knowledge, this work represents the first comprehensive study of the inhomogeneous nonlinear Schrödinger equation with a leading-order focusing energy-critical inhomogeneous nonlinearity and a defocusing perturbation. Our results provide new insights into the interplay between these competing nonlinearities and their influence on the dynamics of solutions.

Theorems & Definitions (48)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Proposition 1.1
• Theorem 1.3
• Remark 1.3
• Remark 1.4
• Theorem 1.4