On the uniqueness of solutions to gauge covariant Poisson equations with   compact Lie algebras
  It is shown, under rather general smoothness conditions on the gauge potential, which takes values in an arbitrary semi-simple compact Lie algebra ${\bf g}$, that if a (${\bf g}$-valued) solution to the gauge covariant Laplace equation exists, which vanishes at spatial infinity, in the cases of 1,2,3,... space dimensions, then the solution is identically zero. This result is also valid if the Lie algebra is merely compact. Consequently, a solution to the gauge covariant Poisson equation is uniquely determined by its asymptotic radial limit at spatial infinity. In the cases of one or two space dimensions a related result is proved, namely that if a solution to the gauge covariant Laplace equation exists, which is unbounded at spatial infinity, but with a certain dimension-dependent condition on the asymptotic growth of its norm, then the solution in question is a covariant constant. 
Christofer Cronstrom