We provide a proof of the following well known fact.
Theorem. There is no
solution to
I found this proof in a paper due to Y.Y. Li published in Commun. Math. Phys. in 1999 [here]. Before deriving the proof, let us recall the following notation known as the sphere mean in the literature. In we denote the integral
by . We call
the average of
on the sphere
of radius
, or sphere mean of a function around the origin. In this context, we simply have
.