Assuming $\Omega \subset \mathbb R^n, (n\ge 3)$ is a domain. Consider the Dirichlet exterior problem \begin{align} &\Delta u = 0 ~~~~x\in \mathbb R^n/\Omega\\ &u|_{\partial \Omega} =1 \tag{1} \end{align} How to show $\lim\limits_{|x|\rightarrow \infty} u(x)=0$ ?
In fact, in my reading book, consider a big ball $B_R(0)$ such that $\Omega \subset B_R(0)$, consider the follow problem \begin{align} &\Delta u = 0 ~~~~x\in B_{R}(0)/\Omega\\ &u|_{\partial \Omega} =1,~~~~~u|_{\partial B_R(0)} =0 \tag{2} \end{align} Then (1) is the limit of (2) as $R\rightarrow \infty$, so $\lim\limits_{|x|\rightarrow \infty} u(x)=0$. But I feel this statement is unconvincing.
The result isn't true regardless of the dimension. Consider for instance the function $u =1$ identically. This is clearly harmonic and satisfies $u=1$ on $\partial \Omega$. Perhaps the authors of your text have build the limit condition into the definition of solution they're using?