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)- \langle M_i(x)D u_i, D u_i \rangle = \lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $
for $ i = 1, \cdots, n $, in a bounded $ C^{1, 1} $ domain $ \Omega\subset \mathbb{R}^N $ with Dirichlet boundary conditions; here $ n\geq 1 $, $ \lambda \in \mathbb{R} $, $ c_{ij}, \, h_i \in L^\infty(\Omega) $, $ c_{ij}\geq 0 $, $ M_i $ satisfies $ 0<\mu_1 I\leq M_i\leq \mu_2 I $, 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.
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Illustration of Theorem 2.4
Illustration of Theorem 2.5 for