We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from $\mathbb R^n$ to $\mathbb S^n$ and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.

中文翻译：

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与保形对数 Sobolev 不等式相关的方程解的分类

我们将方程的所有有限能量解归类为由 Beckner 引起的球面上保形不变对数 Sobolev 不等式的欧拉-拉格朗日方程。我们的证明使用了从 $\mathbb R^n$ 到 $\mathbb S^n$ 的移动球体方法的扩展以及 Li 和 Zhu 的分类结果。在此过程中，我们证明了与对数拉普拉斯算子密切相关的底层算子的小体积最大原理和强最大原理。