I have the following problem with exercise 5 in the book Harmonic funtion theory by Sheldon Axler, Paul Bourdon and Wade Ramey:
Show that if $ n > 2$, then the only harmonic function on $\mathbb{R}^n \cup \{\infty\}$ is identically zero.
Hoping for any hints or proofs to this problem, I am kind of stuck.
In the way you stated the problem, a simple counterexample is the constant function. Maybe you are asking for square-integrable functions? Then use the Kelvin transform to shrink the whole space to the origin. The only function there is the constant. Transforming it back you get a polynomial, which is of course not square-integrable.