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arXiv · 2610.34979

On Brezis' open problem 2.4

Hong-Ge Chen·Yong Liu·Juncheng Wei·Wen Yang
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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.

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Hong-Ge Chen, Yong Liu, Juncheng Wei, Wen Yang. 2026-10-26. On Brezis' open problem 2.4. https://arxiv.org/abs/2610.34979

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