Normalized Standing Waves for the Focusing Inhomogeneous Schr\"odinger Equation with Spatially Growing Nonlinearity

TL;DR

This paper investigates the existence, stability, and instability of ground state standing waves for the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity, extending classical theory to this new setting.

Contribution

It characterizes ground states via variational methods, establishes stability in the subcritical case, and demonstrates instability and blow-up in critical and supercritical regimes.

Findings

Existence of normalized ground states in the subcritical regime.

Orbital stability of ground states in the subcritical case.

Strong instability and finite-time blow-up in critical and supercritical regimes.

Abstract

We study the focusing inhomogeneous nonlinear Schr\"odinger equation $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ 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 Schr\"odinger equations to the case of spatially growing inhomogeneous nonlinearities.