Systems of Bellman equations to stochastic differential games with non-compact coupling

We consider a class of non-linear partial differential systems like - div (a(x)∇u_ν) + λu_ν = H_ν(x, Du), with applications for the solution of stochastic differential games with N players, where N is an arbitrary but positive number. The Hamiltonian H of the non-linear system satisfies a quadratic growth condition in Du and contains interactions between the players in the form of non-compact coupling terms ∇u_i ∇u_j. A L^∞ ∩ H ¹-estimate and regularity results are shown, mainly in two-dimensional space. The coupling arises from cyclic non-market interaction of the control variables.