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I have proved the first part of the problem below, but I don't know how to proof the second one (I must admit I cannot even find the relation between both assertions). The statement of the problem is as follows:

(i) Let $u\in C(\mathbb{R}^n)$ a subharmonic function, $u\leq 0$ in $\mathbb{R}^2$. Prove that in that case u is a constant function.

(ii) Prove that in $\mathbb{R}^n$ being $n\geq 3$, there exist subharmonic functions $u\in C^\infty (\mathbb{R}^n)$ such that $|| u ||_{L^\infty (\mathbb{R}^n)}<\infty$.

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(ii) should state "nonconstant subharmonic", otherwise the trivial constant function works for part (ii).

Recall the fundamental solution of Laplace's equation is

$$u(x) = |x|^{2-n}$$

up to a constant, for $n\geq 3$ (so $\Delta u =0$ for $x\neq 0$). The idea is to smooth out the singularity while keeping the function subharmonic. Try something like

$$u(x) = \frac{1}{\sqrt{|x|^2+\epsilon^2}}$$

for a small positive number $\epsilon>0$.

The relation between (i) and (ii) is that in (i), you showed that every subharmonic function in $n=2$ variables that is bounded above is constant (the bound can be anything $u\leq C$ for a constant $C$). In (ii) you show that this does not hold in higher dimensions $n\geq 3$, that is there exist bounded subharmonic functions in higher dimensions that are not constant.