I'd like to know more about,
$$\lambda u = \nabla ^2 u$$
for unbounded domains (particularly the exterior of a disk in $\mathbb{R}^2$ or ball in $\mathbb{R}^3$), but have had a hard time finding references online. Can anyone suggest a website or book that covers this in depth? i.e. How to obtain solutions, proofs, important corollaries/theorems, etc.