We consider the equation Δu=wf ′(u)
on a symmetric bounded domain in Rn
with Dirichlet boundary conditions.
Here w is a positive function or measure
that is invariant under the (Euclidean) symmetries of the domain.
We focus on solutions u that are positive and/or
have a low Morse index.
Our results are concerned with the existence of non-symmetric
solutions and the non-existence of symmetric solutions.
In particular, we construct a solution u for the disk in R2
that has index 2 and whose modulus |u|
has only one reflection symmetry.