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

Abstract

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \({\mathbb R^n}\) . The solutions take values in ordered Banach spaces E, e.g. \({E=\mathbb R^N}\) ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.