Normalized Standing Waves for the Focusing Inhomogeneous Schrödinger Equation with Spatially Growing Nonlinearity

TL;DR

This work analyzes the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u $$ and $b>0$, developing a variational Nehari framework to characterize ground states. It proves a sharp threshold dynamics: ground states minimize the action on the Nehari manifold, yield a gauge-invariant orbit $\{e^{i\theta}Q_\omega\}$, and separate the dynamics into global existence or finite-time blow-up below the ground-state energy via a potential-well structure. The paper also constructs normalized standing waves in the $L^2$-subcritical regime, establishing their existence, compactness, and orbital stability, while proving strong instability in the critical/supercritical regimes. Overall, it extends classical NLS stability theory to spatially growing inhomogeneous nonlinearities by leveraging radial compactness and tailored variational methods, and it provides a coherent threshold framework for both unconstrained and mass-constrained standing waves.

Abstract

We study the focusing inhomogeneous nonlinear Schrödinger equation $$ i\partial_t u + Δu = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity, standard compactness arguments do not apply and new difficulties arise. We first characterize ground state standing waves via a variational approach on the Nehari manifold and we establish some sharp stability and instability properties. In the $L^2$-subcritical regime, we prove the existence of normalized ground states by solving a constrained energy minimization problem in the radial energy space, and we show that the resulting set of minimizers is orbitally stable under the flow. In contrast, in the $L^2$-critical and supercritical regimes, ground state standing waves are shown to be strongly unstable by finite-time blow-up. Our results extend classical stability and instability theory for nonlinear Schrödinger equations to the case of spatially growing inhomogeneous nonlinearities.

Theorems & Definitions (33)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Definition 1.1
• Remark 1.3
• Theorem 1.3
• Remark 1.4
• Proposition 1.4
• Remark 1.5