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
Proof. We derive from the Jensen inequality that
It follows that satisfies
in , namely,
We derive from the above that
for all , i.e. is monotone increasing. Consequently,
for all . In turn we have
for all . It now follows that
diverges. A contradiction.