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In recent years there has been intense interest in the vanishing discountproblem for Hamilton-Jacobi equations. In the case of the scalar equation, B.Ziliotto has recently given an example of the Hamilton-Jacobi equation havingnon-convex Hamiltonian in the gradient variable, for which the full convergenceof the solutions does not hold as the discount factor tends to zero. We givehere an explicit example of nonlinear monotone systems of Hamilton-Jacobiequations having convex Hamiltonians in the gradient variable, for which thefull convergence of the solutions fails as the discount factor goes to zero.

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence