Sharp global existence and orbital stability of standing waves for the Schrödinger-Hartree equation with partial confinement

Feiyan Lei; Hui Jian; Feiyan Lei; Hui Jian

doi:10.3934/math.20251073

AIMS Mathematics

2025, Volume 10, Issue 10: 24208-24239. doi: 10.3934/math.20251073

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Sharp global existence and orbital stability of standing waves for the Schrödinger-Hartree equation with partial confinement

Feiyan Lei ,
Hui Jian ^,

School of Science, East China Jiaotong University, Nanchang 330013, China

Received: 19 August 2025 Revised: 03 October 2025 Accepted: 16 October 2025 Published: 23 October 2025
MSC : 35A01, 35A15, 35B44, 35Q55

This study is concerned with the sharp criterion of global existence and the orbital stability of standing waves for the Hartree equation in presence of a partial confinement. Using the scaling technique and constructing cross-invariant sets, we first derive the sharp threshold for global existence and blow-up of the solution in both $ L^{2} $-critical and $ L^{2} $-supercritical settings. Then, by taking advantage of the profile decomposition technique and concentration compact arguments together with variational methods, we explore the existence of normalized standing waves and show that these standing waves are orbitally stable in the $ L^{2} $-subcritical, $ L^{2} $-critical, and supercritical cases. Our conclusions complement and compensate some previous results.
- Hartree equation,
- partial confinement,
- global existence,
- sharp threshold,
- stability
Citation: Feiyan Lei, Hui Jian. Sharp global existence and orbital stability of standing waves for the Schrödinger-Hartree equation with partial confinement[J]. AIMS Mathematics, 2025, 10(10): 24208-24239. doi: 10.3934/math.20251073

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Abstract

This study is concerned with the sharp criterion of global existence and the orbital stability of standing waves for the Hartree equation in presence of a partial confinement. Using the scaling technique and constructing cross-invariant sets, we first derive the sharp threshold for global existence and blow-up of the solution in both $ L^{2} $-critical and $ L^{2} $-supercritical settings. Then, by taking advantage of the profile decomposition technique and concentration compact arguments together with variational methods, we explore the existence of normalized standing waves and show that these standing waves are orbitally stable in the $ L^{2} $-subcritical, $ L^{2} $-critical, and supercritical cases. Our conclusions complement and compensate some previous results.

References

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