Quantitative analysis of ground states for the fractional logarithmic Schrödinger equation
Abstract
Let N≥1 and 0<s<1. We study positive ground states of the fractional logarithmic Schrödinger equation (−Δ)sQ=Qlog Q in ℝN. We prove that for every N≥1 and 0<s<1, the positive ground state is unique up to translations and nondegenerate. More precisely, for the linearized operator LQ=(−Δ)s−1−log Q, it holds that ker LQ=span{∂x1Q,⋯,∂xNQ}. A main difficulty is that the potential −1−log Q is unbounded in ℝN, which prevents a direct application of the available radial oscillation theory for fractional Schrödinger operators with bounded potentials. We overcome this difficulty by a bounded-potential approximation. Using also the fact that the associated quadratic form has Morse index one and an angular decomposition, we obtain the nondegeneracy. Based on the isolation of logarithmic ground states and the uniqueness theory for the fractional power equation, we prove uniqueness by a variational approximation with subcritical power nonlinearities. As an application, we establish sharp fractional logarithmic Sobolev inequalities and characterize all cases of equality.
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BibTeX
@misc{an2026quantitative,
title = {{Quantitative analysis of ground states for the fractional logarithmic Schrödinger equation}},
author = {Xiaoming An and Shuangjie Peng and Fulin Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10254},
eprint = {2609.10254},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10254},
}
RIS
TY - UNPB
TI - Quantitative analysis of ground states for the fractional logarithmic Schrödinger equation
AU - Xiaoming An
AU - Shuangjie Peng
AU - Fulin Zhong
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10254
UR - https://arxiv.org/abs/2609.10254
AB - Let N≥1 and 0<s<1. We study positive ground states of the fractional logarithmic Schrödinger equation (−Δ)^sQ=Qlog Q in ℝ^N. We prove that for every N≥1 and 0<s<1, the positive ground state is unique up to translations and nondegenerate. More precisely, for the linearized operator L_Q=(−Δ)^s−1−log Q, it holds that ker L_Q=span{∂_x_1Q,⋯,∂_x_NQ}. A main difficulty is that the potential −1−log Q is unbounded in ℝ^N, which prevents a direct application of the available radial oscillation theory for fractional Schrödinger operators with bounded potentials. We overcome this difficulty by a bounded-potential approximation. Using also the fact that the associated quadratic form has Morse index one and an angular decomposition, we obtain the nondegeneracy. Based on the isolation of logarithmic ground states and the uniqueness theory for the fractional power equation, we prove uniqueness by a variational approximation with subcritical power nonlinearities. As an application, we establish sharp fractional logarithmic Sobolev inequalities and characterize all cases of equality.
ER -