  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.
