ON THE VANISHING DISCOUNT PROBLEM FROM THE NEGATIVE DIRECTION

It has been proved in [7] that the unique viscosity solution of lambda u(lambda) + H(x,d(x)u(lambda)) = c(H) in M, (*) uniformly converges, for lambda -> 0(+), to a specific solution up of the critical equation H(x, d(x)u) = c(H) in M, where M is a closed and connected Riemannian manifold and c(H) is the critical value. In this note, we consider the same problem for lambda -> 0(-). In this case, viscosity solutions of equation (*) are not unique, in general, so we focus on the asymptotics of the minimal solution u(lambda)(-) , of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the u(lambda)(-), also converges to up as lambda -> 0(-) . Furthermore, we exhibit an example of H for which equation (*) admits a unique solution for lambda < 0 as well.

Automated summary

This paper discusses the convergence behavior of the solutions of a critical equation as lambda approaches 0 in different directions. It focuses on the asymptotics of the solutions and presents an example where the equation may have a unique solution under certain conditions.