Abstract

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as -F-i(x, u(i), Du(i), D(2)u(i)) - M-i(x) Du(i), Du(i)> = lambda c(i1)(x)u(1 )+...+ lambda c(in)(x) u(n) + h(i)(x), for i = 1, ..., n, in a bounded C-1,C-1 domain Omega subset of R-N with Dirichlet boundary conditions; here n >= 1, lambda is an element of R, c(ij) , h(i) is an element of L-infinity (Omega), c(ij) >= 0, M-i , satisfies 0 mu I-1 = M-i = mu I-2, and F-i is an uniformly elliptic Isaacs operator. We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.