I'm having some trouble with the following:
Let $u:\mathbb{R}^2\setminus\{0\}\to [0,\infty)$ be a harmonic function. Show that $u$ is constant.
I have seen different proves for this. However, we are supposed to do it by using proper barriers to show $u\geq \min\{u(x):|x|=1\}$ first.
Let us consider u = c(constant) then u is harmonic function also this implies that u is analytic function and holomorphic so now we are defining u on some bounded interval and hence by applying L-theorem the function must be constant and also it leaves at most one point and define uncountable no. of points thats(picard theorem as 0 is elemented ) thats why u is constant.