  I investigate stable equilibria of bodies in potential fields satisfying a
generalized Poisson equation: divergence[m(grad phi) grad phi]= source density.
This describes diverse systems such as nonlinear dielectrics, certain flow
problems, magnets, and superconductors in nonlinear magnetic media; equilibria
of forced soap films; and equilibria in certain nonlinear field theories such
as Born-Infeld electromagnetism. Earnshaw's theorem, totally barring stable
equilibria in the linear case, breaks down. While it is still impossible to
suspend a test, point charge or dipole, one can suspend point bodies of finite
charge, or extended test-charge bodies. I examine circumstances under which
this can be done, using limits and special cases. I also consider the analogue
of magnetic trapping of neutral (dipolar) particles.
