Nonuniqueness of Dirichlet data without control at infinity

Unbounded-domain Dirichlet problems can need a condition at infinity. On all of ${\mathbb{R}}^n$, constant functions have identical empty boundary data and are distinct finite-energy stationary points of $\frac{1}{2}\int |Du{|}^2$. With nonempty boundary, on the exterior of the unit ball in dimension at least three, $1-|x{|}^{2-n}$ is a nonzero harmonic function vanishing on the sphere, with finite Dirichlet energy and a nonzero limit at infinity. It is stationary for compactly supported variations. The permitted class of variations and the behaviour at infinity must be specified for a global variational uniqueness claim.