PreprintarXiv licencearXiv
On Brezis' open problem 2.4
Abstract
We show the existence of a ``nonradial" smooth entire solution of the Ginzburg--Landau equation with degree q∈{8,9} and modulus tending to one at infinity. This answers Brezis' Open Problem 2.4 negatively. To construct the counterexample, we vary a cone parameter q ∈ [8, 9] and keeping the degree on the cone equal to one. Several key ingredients are needed. First, we need to compute the Morse index of radial vortex solutions at q∈ {8, 9}. Second, we need to show compactness. Finally, Leray-Schauder degree theory and a homotopy argument yield the desired existence result.
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BibTeX
@misc{chen2026brezis,
title = {{On Brezis' open problem 2.4}},
author = {Hong-Ge Chen and Yong Liu and Juncheng Wei and Wen Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34979},
eprint = {2610.34979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34979},
}
RIS
TY - UNPB
TI - On Brezis' open problem 2.4
AU - Hong-Ge Chen
AU - Yong Liu
AU - Juncheng Wei
AU - Wen Yang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34979
UR - https://arxiv.org/abs/2610.34979
AB - We show the existence of a ``nonradial" smooth entire solution of the Ginzburg--Landau equation with degree q∈{8,9} and modulus tending to one at infinity. This answers Brezis' Open Problem 2.4 negatively. To construct the counterexample, we vary a cone parameter q ∈ [8, 9] and keeping the degree on the cone equal to one. Several key ingredients are needed. First, we need to compute the Morse index of radial vortex solutions at q∈ {8, 9}. Second, we need to show compactness. Finally, Leray-Schauder degree theory and a homotopy argument yield the desired existence result.
ER -