On Brezis' open problem 2.4

We show the existence of a ``nonradial" smooth entire solution of the Ginzburg--Landau equation with degree $q\in\{8,9\}$ and modulus tending to one at infinity. This answers Brezis' Open Problem 2.4 negatively. To construct the counterexample, we vary a cone parameter $q \in [8, 9]$ 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 $q\in \{8, 9\}$. Second, we need to show compactness. Finally, Leray-Schauder degree theory and a homotopy argument yield the desired existence result.

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-10-26
DOI
https://doi.org/10.48550/arxiv.2610.34979
Primary Topic
Nonlinear Partial Differential Equations
Type
preprint
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preprint

On Brezis' open problem 2.4

Juncheng Wei, Hong-Ge Chen, Wen Yang, Yong Liu
arXiv (Cornell University)
Nonlinear Partial Differential Equations
preprint

On Brezis' open problem 2.4

Juncheng Wei, Hong-Ge Chen, Wen Yang, Yong Liu
preprint en

Abstract

We show the existence of a ``nonradial" smooth entire solution of the Ginzburg--Landau equation with degree $q\in\{8,9\}$ and modulus tending to one at infinity. This answers Brezis' Open Problem 2.4 negatively. To construct the counterexample, we vary a cone parameter $q \in [8, 9]$ 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 $q\in \{8, 9\}$. Second, we need to show compactness. Finally, Leray-Schauder degree theory and a homotopy argument yield the desired existence result.

arXiv (Cornell University)
Nonlinear Partial Differential Equations
DOIOpen Access PDFOpenAlex
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On Brezis' open problem 2.4 — Juncheng Wei, Hong-Ge Chen, et al. · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS