Nonradial stable solutions near the Joseph--Lundgren threshold

Shibing Chen, Yong Liu, Juncheng Wei, Wen Yang

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.

Disclosure

“he Macao SAR (FDCT, Grant No. 0070/2024/RIA1), the Multi-Year Research Grants of the University of Macau (Grant No. MYRG-GRG2025-00051-FST), and the University of Macau Development Foundation (Grant No. TISF/2025/006/FST). The authors used OpenAI models as assistive tools in preparing this manuscript. Mathematical arguments, proofs, and computations in the manuscript were verified and finalized by the authors. References [1] T. Aubin,”
PDF page 40

Classification: Drafting limited passages
Multiplier: 5
Verified: 2026-08-23T05:09:08.748366Z

Structural counts

Pages 43 pdf

Theorems 6 source

Lemmas 25 source

Propositions 5 source

Corollaries 1 source

Definitions 1 source

Displayed equations 393 source

Bibliography entries 54 source

Appendix pages 6 estimated

Count notes

Source counts use the expanded primary TeX file Nonradial-stable.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.