Classification of solutions of an equation related to a conformal log Sobolev inequality

TL;DR

This paper classifies all finite energy solutions to a conformally invariant logarithmic Sobolev inequality-related equation on the sphere, extending the method of moving spheres and establishing new maximum principles.

Contribution

It extends the method of moving spheres to the sphere and classifies solutions of a key conformal equation, linking to the logarithmic Laplacian.

Findings

Complete classification of solutions on the sphere.

Development of maximum principles for the logarithmic Laplacian.

Extension of moving spheres method to spherical setting.

Abstract

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.