AI 中文总结
该研究针对超临界Lane-Emden方程，在第一Joseph-Lundgren区间构造了指数接近上端点的非径向稳定整体解，否定了Chan和Wei的径向性猜想，是Emden-Fowler方程非径向稳定解的首个非平凡例子。
AI 中文摘要
我们研究第一Joseph-Lundgren区间中超临界Lane-Emden方程的正稳定解。对于一系列维度，我们构造了指数接近该区间上端点的非径向稳定整体解，这否定了Chan和Wei的径向性猜想。该构造始于球面上的光滑正非常数解，该解通过将极帽与内颈匹配得到。最低移位特征值的精确展开证明所得奇异锥是严格稳定的。最后，极小解与重标度论证将该锥替换为光滑稳定整体解，同时保留其球变分。据我们所知，这是Emden-Fowler方程非径向稳定解的首个非平凡例子。
英文摘要
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.
