  We investigate the existence of invariant measures for self-stabilizing
diffusions. These stochastic processes represent roughly the behavior of some
Brownian particle moving in a double-well landscape and attracted by its own
law. This specific self-interaction leads to nonlinear stochastic differential
equations and permits to point out singular phenomenons like non uniqueness of
associated stationary measures. The existence of several invariant measures is
essentially based on the non convex environment and requires generalized
Laplace's method approximations.

Probability