Some symmetric boundary value problems and non-symmetric solutions

We consider the equation −Δu=wf′(u)−Δu=wf′(u) on a symmetric bounded domain in RnRn 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 R2R2 that has index 2 and whose modulus |u||u| has only one reflection symmetry. We also provide a corrected proof of [12, Theorem 1].