Problem
Solve $\Delta u=0$ on $\Omega$, where $\Omega=\{x : \|x\|>1\}$. The conditions are $u=1$ on the boundary of $\Omega$, and $\lim_{x\to\infty}u(x)=0$.
Context
The domain here is the exterior of unit ball in $\mathbb R^n$. If it was the interior, then the only harmonic function with $u=1$ on the boundary would be the constant one, $u\equiv 1$.
But here, $u\equiv 1$ is not an acceptable solution, as it does not satisfy $\lim_{x\to\infty}u(x)=0$.