A subharmonic function

Question

When $u=u(x)>0$ is a smooth non-constant function in $\mathbb{R^n}$ and is subharmonic in $\mathbb{R^n}$, i.e.

$u\geq0$ in $\mathbb{R^n}$,

can we conclude that $u$ is unbounded in $\mathbb{R^n}$ ?

Answer 1

No. For example, in $\mathbb R^3$ the function $u(x)=\max(1,2-|x|^{-1})$ is subharmonic. You can also make it smooth by mollification.