Non-symmetric low-index solutions for a symmetric boundary value problem

We consider the equation −Δu=wu3−Δu=wu3 on a square domain in R2R2, with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equation is shown to have a solution u, with Morse index 2, that is neither symmetric nor antisymmetric with respect to any nontrivial symmetry of the square. Part of our proof is computer-assisted. An analogous result is proved for index 1.