Symmetry breaking solutions of nonlinear elliptic systems

We consider nonlinear elliptic systems with Dirichlet boundary condition on a bounded domain in R-N which is invariant with respect to the action of some group G of orthogonal transformations. For every subgroup K of G we give a, simple criterion for the existence of infinitely many solutions which are K-invariant but not G-invariant. We include a detailed discussion of the case N = 3.