Ground states and sharp logarithmic Sobolev inequalities for logarithmic Schrödinger equations with partial confinement
Abstract
We study the logarithmic Schrödinger equation with an arbitrary positive semidefinite quadratic potential and a periodic potential in the unconfined directions. The main variational difficulty is to turn a constrained minimizer into a critical point when the energy is extended-valued and not of class C1 on the natural Hilbert space. We establish an abstract principle for convex lower semicontinuous energies with a smooth perturbation, allowing a scaling law for the full energy without requiring homogeneity of its convex part. Combined with a profile decomposition under integer translations, this principle yields a positive ground state for every rank of the quadratic form. When the periodic potential vanishes, we prove a sharp anisotropic logarithmic Sobolev inequality, determine its optimal constant and all equality cases, and classify positive ground states as explicit anisotropic Gaussons, unique up to translations along the kernel of the quadratic form. When the periodic component is constant, we also establish existence and classify all normalized ground states at every prescribed positive mass, with explicit formulas for the minimum energy and the associated Lagrange multiplier.
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BibTeX
@misc{li2026ground,
title = {{Ground states and sharp logarithmic Sobolev inequalities for logarithmic Schrödinger equations with partial confinement}},
author = {Baihong Li and Jarosław Mederski and Xiangjian Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13887},
eprint = {2610.13887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13887},
}
RIS
TY - UNPB
TI - Ground states and sharp logarithmic Sobolev inequalities for logarithmic Schrödinger equations with partial confinement
AU - Baihong Li
AU - Jarosław Mederski
AU - Xiangjian Zeng
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13887
UR - https://arxiv.org/abs/2610.13887
AB - We study the logarithmic Schrödinger equation with an arbitrary positive semidefinite quadratic potential and a periodic potential in the unconfined directions. The main variational difficulty is to turn a constrained minimizer into a critical point when the energy is extended-valued and not of class C^1 on the natural Hilbert space. We establish an abstract principle for convex lower semicontinuous energies with a smooth perturbation, allowing a scaling law for the full energy without requiring homogeneity of its convex part. Combined with a profile decomposition under integer translations, this principle yields a positive ground state for every rank of the quadratic form. When the periodic potential vanishes, we prove a sharp anisotropic logarithmic Sobolev inequality, determine its optimal constant and all equality cases, and classify positive ground states as explicit anisotropic Gaussons, unique up to translations along the kernel of the quadratic form. When the periodic component is constant, we also establish existence and classify all normalized ground states at every prescribed positive mass, with explicit formulas for the minimum energy and the associated Lagrange multiplier.
ER -