Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces (Q661309)

For problems \(-\Delta u = f(x,u)\) in bounded symmetric domains \(\Omega \subset \mathbb{R}^n\), \(n \geq 1\), \(u = 0\) on \(\partial \Omega\), it is natural to ask what is the symmetry of solutions. For \(\Omega\) symmetric with respect to a hyperplane (say \(x_1 = 0\)), the authors find conditions on \(f\) with values in \(E = \mathbb{R}^N\) (or in an infinite dimensional Banach space \(E\)), ordered by a cone \(K \subset E\), such that solutions will have the same reflection symmetry with respect to \(x_1 = 0\).