Some symmetric boundary value problems and non-symmetric solutions (Q2342731)

Let \(\Omega \subset R^n\) be a symmetric bounded domain with Lipschitz boundary. The paper studies in \(H^1(\Omega )\) the nonlinear Dirichlet problem \(-\Delta u=wf'(u)\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), where \(w\) is a symmetric nonnegative function or a symmetric measure. The authors concentrate on solutions that are positive or have a low Morse index. The paper shows that even if the problem is symmetric, a solution might be nonsymmetric.