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

Abstract

We study the focusing inhomogeneous nonlinear Schrödinger equation $$i{\partial }_{t}u+\Delta u=-|x{|}^b|u{|}^{p-1}u,(t,x)\in (0,\infty )\times {\mathbb{R}}^N,$$ 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.