Normalized Standing Waves for the Focusing Inhomogeneous Schrödinger Equation with Spatially Growing Nonlinearity
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arXiv
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| Format: | Preprint |
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2026
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| author | Majdoub, Mohamed Saanouni, Tarek |
| author_facet | Majdoub, Mohamed Saanouni, Tarek |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07505 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normalized Standing Waves for the Focusing Inhomogeneous Schrödinger Equation with Spatially Growing Nonlinearity Majdoub, Mohamed Saanouni, Tarek Analysis of PDEs 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. |
| title | Normalized Standing Waves for the Focusing Inhomogeneous Schrödinger Equation with Spatially Growing Nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.07505 |