Already the fact that $u$ is a tempered distribution and is weakly harmonic implies that $u$ is a polynomial. Then observe that the only polynomial in $L^p({\mathbb R}^n)$ is zero. Also, you do not need to talk about "almost everywhere" because weakly harmonic functions are smooth (in fact analytic).
timur
- 3.3k
- 1
- 36
- 42