ABSTRACT
This chapter discusses whether positive solutions of a second order elliptic nonlinear partial differential equation with a certain symmetry (e.g., spherical symmetry) exhibit the same symmetry. For a large class of nonlinearities, it shows that positive solutions respect the symmetry of the equation. The chapter treats equations which are classical approximations to quantum or statistical mechanics systems. A symmetry may not be broken at the classical level, but broken only after quantum or thermodynamic effects are taken into account. The chapter also provides positive solutions of the second order elliptic nonlinear partial differential equation in bounded domains.
