-Assume $u$ is a harmonic function on $\mathbb{R}^3$, and assume u(x,y,z)=1+x when $x^2+y^2+z^2=1$. What is the value $u(0,0,0)$?
I am in doubt between $1$ and $0$, could it be both?
-Let $V(r)$ be a radial harmonic function in $\mathbb{R}^3$; which is the ODE for V?
My answer: $$V'' + \frac 2 r V' = 0, \quad \text{for} \quad r>0. $$
are they correct?
Your answer to the second part is correct. For the firwt part, the value of a harmonic function at the center of the surface is equal to the average of its value over that surface. By symmetry, that average value is 1, so that's your answer.