  We consider in this paper elliptic equations which are perturbations of
Laplace's equation by a compactly supported potential. We show that in
dimension greater than three for a wide class of potentials all the solutions
are globally minimising. However, in dimension two the situation is different.
We show that for radially symmetric potentials there always exist solutions
which are not locally minimal unless the potential vanishes identically. We
discuss the relations of this with the so-called Hopf rigidity phenomenon.
