Some symmetric boundary value problems and non-symmetric solutions

We consider the equation -Delta u = wf'(u) on a symmetric bounded domain in R-n 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 R-2 that has index 2 and whose modulus vertical bar u vertical bar has only one reflection symmetry. We also provide a corrected proof of [12, Theorem 1]. (C) 2015 Elsevier Inc. All rights reserved.