Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space

Abstract

We study classical solutions of elliptic systems in the half-space and provide sufficient conditions for having symmetry (or proportionality) of components, i.e. u=Kv with K>0, which then reduces the system to the scalar case. Under a natural structure condition on the nonlinearities, we show that solutions with sublinear growth, hence in particular bounded solutions, are symmetric. Noncooperative, nonvariational systems as well as supercritical nonlinearities can be covered. We also give an instance of our proportionality results without growth restriction on the solutions. As a consequence, we obtain new Liouville-type theorems in the half-space, as well as a priori estimates and existence results for related Dirichlet problems. Our proofs are based on a maximum principle, on the properties of suitable half-spherical means, on a rigidity result for superharmonic functions and on nonexistence of solution for scalar inequalities on the half-space.