We are given domain $\Omega=\{x\in R^n:|x|>1\}$ and let $u \in C^2(\Omega)\cap C^1(\overline \Omega)$ satisfy $\bigtriangleup u=0, \quad \text{in} \quad \Omega$. $$$$ Assume $\lim_{|x| \to \infty}|u(x)|=0$. (1)$$$$ Prove that $\sup_{\Omega}|u|=\max_{|x|=1}|u|$ $$$$ Is assumption (1) necessary? Construct a counterexample in $R^{2}$.
The mean value theorem implies that the maximum cannot occur in the interior of the boundary and assumption 1 implies that the maximum must be attained.
Any help solving this is greatly appreciated!