  In recent years, there have been many contributions to the vanishing discount
problem for Hamilton-Jacobi equations. In the case of the scalar equation, B.
Ziliotto [Convergence of the solutions of the discounted Hamilton-Jacobi
equation: a counterexample. J. Math. Pures Appl. (9) 128 (2019), 330-338] has
shown an example of the Hamilton-Jacobi equation having non-convex Hamiltonian
in the gradient variable, for which the full convergence of the solutions does
not hold as the discount factor tends to zero. We give an example of the
nonlinear monotone system of Hamilton-Jacobi equations having convex
Hamiltonians in the gradient variable, for which the whole family convergence
of the solutions does not hold.
