Is there an analogue of Earnshaw’s theorem for inverse cube force laws? Earnshaw’s theorem states that point charges cannot be held in electrostatic equilibrium by an electric field. Is there an analogue of this for inverse cube force laws?
 A: I had a dilemma - whether to give a long or a short answer. The long one would contain a general discussion of the proof of Earnshaw's theorem and of what it becomes for a different force law. The short one gives only the results. I've decided for the latter. If you are interested in a
deeper discussion please put a specific question.
Earnshaw's theorem rests on electrostatic potential satisfying (in vacuum) Laplace's equation
$$\nabla^2 V = 0.\tag1$$
If potential of a point charge is $\propto 1/r$ then eq. (1) follows for whatever distribution of charges, in any region devoid of charge. Then it's easy to see that $V$ can have neither local maxima nor local minima, so that stable equilibrium for a test charge is ruled out.
If potential goes as $1/r^2$ (or any other power of $r$) Laplace's equation doesn't hold, but this doesn't logically imply that stable equilibrium is allowed - a more thorough discussion is required. But it's not difficult to find positive examples. 
E.g. equal charges at vertices of a regular octahedron give stable equilibrium for a charge of same sign placed in the centre.
