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arXiv · 2608.10501

Nonradial stable solutions near the Joseph--Lundgren threshold

Shibing Chen·Yong Liu·Juncheng Wei·Wen Yang
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Abstract

We study positive stable solutions of the supercritical Lane--Emden equation in the first Joseph--Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is obtained by matching a polar cap to an inner neck. A sharp expansion of the lowest shifted eigenvalue proves that the resulting singular cone is strictly stable. Finally, a minimal-solution and rescaling argument replaces the cone by a smooth stable entire solution while preserving its sphere variation. To the best of our knowledge, this is the first nontrivial example of nonradial stable solutions for the Emden-Fowler equation.

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Shibing Chen, Yong Liu, Juncheng Wei, Wen Yang. 2026-08-11. Nonradial stable solutions near the Joseph--Lundgren threshold. https://arxiv.org/abs/2608.10501

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