We can't use the ordinary maximum principle for harmonic functions since our domain is unbounded. We hope to use what we would expect the maximum principle would be for this domain, but that requires proof.
Lemma: For the domain , if then .
Proof of Lemma: Consider the domain and the function . Since is the fundamental solution to Laplaces equation, then clearly is harmonic and one can easily verify that on .
Then since the domain is bounded, we can use the ordinary maximum principle and say that
We know that . Now if is sufficiently large then we can guarantee that . Sending gives us . Now send and we get which proves the lemma.
If we replace with then by the same methods we can prove a minimum principle for our domain .
Now suppose are two bounded solutions to the Dirichlet problem. Let . Then solves in and on . Then by our maximum principle lemma, . Similarly by the minimum principle, . This implies , i.e. , which proves that bounded solutions to this Dirichlet problem are unique.
Now when we can have more than one bounded solution to the Dirichlet problem on . Suppose is one such bounded solution. Recall that is the fundamental solution to Laplace's equation in dimension 3. Therefore is also harmonic and one can verify that on . It is also easy to verify that is also bounded on . Therefore both are distinct, bounded solutions. Therefore solution is not unique.
How can we get a unique solution? Recall that solves in and on . Integration by parts shows that
So if we imposed then we would have which implies that is constant. But the boundary conditions tell us that . In other words, then the solution would be unique.