A Maximum Principle for an Elliptic System and Applications to Semilinear Problems

The Dirichlet problem in a bounded region for elliptic systems of the form \[ ( * ) - Δ u = f(x,u) - v, - Δ v = δ u - γ v\] is studied. For the question of existence of positive solutions the key ingredient is a maximum principle for a linear elliptic system associated with (*). A priori bounds for the solutions of (*). are proved under various types of growth conditions on f. Variational methods are used to establish the existence of pairs of solutions for (*).