I have a unit ball centred at the origin and I know that $u: \mathbb B\rightarrow\mathbb R$ is harmonic and that 1/u is harmonic on $\mathbb B$. I need help proving u is a constant for u(x) $>0$.
Any help will be appreciated, I have spent ages trying to understand this. My approach was to find $\nabla$ of u and 1/u and take a sum somewhere but I kept getting confused.
On
If both are harmonic, the mean value property holds for both. Using Jensen with the strictly convex function $t\mapsto\frac1t$ (for positive $t$) we have $$\frac{1}{u(0)}=\frac{1}{|B|}\int\limits_B \frac{1}{ u(z)}dz\geq \frac{1}{\frac{1}{|B|}\int\limits_B u(z)dz}=\frac{1}{u(0)}$$ and unless $u$ is constant the inequality is strict, which is impossible.
In any case the simplest or most direct way seems to be computing the laplacian, as suggested by zhw..
EDIT: to avoid integrability issues you should take both integrals on a slightly smaller ball.
Hint: $\Delta (uv) = (\Delta u)v + 2 (\nabla u \cdot \nabla v) + u(\Delta v).$