The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 2: Nonlinear coupling

Abstract: We study the vanishing discount problem for a nonlinear monotone system of Hamilton-Jacobi equations. This continues the first author's investigation on the vanishing discount problem for a monotone system of Hamilton-Jacobi equations. As in Part 1, we introduce by the convex duality Mather measures and their analogues for the system, which we call respectively Mather and Green-Poisson measures, and prove a convergence theorem for the vanishing discount problem. Moreover, we establish an existence result for the ergodic problem.