Harnack's Inequality for Cooperative Weakly Coupled Elliptic Systems

Abstract

We consider cooperative, uniformly elliptic systems, with bounded coefficients and coupling in the zeroth-order terms. We establish two analogues of Harnack's inequality for this class of systems. A weak version is obtained under fairly general conditions, while imposing an irreducibility condition on the coupling coefficients we obtain a stronger version of the inequality. This irreducibility condition is also necessary for the existience of a Harnack constant for this class of systems. A Harnack inequality is also obtained for a class of superharmonic functions.