Construction of two-bubble solutions for the energy-critical NLS in dimension 6
Abstract
We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in space dimension N = 6. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate e−|t|. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension N ≥ 7. The six-dimension case presents specific difficulties, as the ground state does not belong to Ḣ−1. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.
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BibTeX
@misc{jendrej2026construction,
title = {{Construction of two-bubble solutions for the energy-critical NLS in dimension 6}},
author = {Jacek Jendrej and Xuemei Li and Guixiang Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16186},
eprint = {2608.16186},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16186},
}
RIS
TY - UNPB
TI - Construction of two-bubble solutions for the energy-critical NLS in dimension 6
AU - Jacek Jendrej
AU - Xuemei Li
AU - Guixiang Xu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.16186
UR - https://arxiv.org/abs/2608.16186
AB - We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in space dimension N = 6. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate e^−|t|. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension N ≥ 7. The six-dimension case presents specific difficulties, as the ground state does not belong to Ḣ^−1. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.
ER -