Selection of critical solutions in non-monotone vanishing discount problems

Abstract

We study a generalized vanishing discount problem for Hamilton--Jacobi equations without assuming monotonicity either pointwise or in the averaged sense with respect to all Mather measures. Specifically, we consider \[ λa(x)u(x)+H\big(x,Du(x)\big)=c_0, \] and assume that, for every sufficiently small $λ>0$, this equation admits a strict subsolution bounded from below uniformly with respect to $λ$. We prove that, under a smallness assumption on the Mather quotient, the maximal viscosity solution converges uniformly as $λ\to 0^+$ under a sign condition on the discount coefficient $a(x)$ determined by a prescribed part of the Aubry set. The limit is characterized explicitly in terms of the Peierls barrier and the selected Mather measures. As an application, the elementary solution of the critical Hamilton--Jacobi equation \[ H\big(x,Du(x)\big)=c_0 \] associated with any isolated static class can be realized as the vanishing discount limit. This provides the first mechanism for selecting multiple critical solutions in vanishing discount problems beyond the classical monotonicity framework, showing that different families of Mather measures yield different limiting critical solutions. We finally propose a variant of the discounted problems for which the strict subsolution hypothesis is automatically verified and hence applies to more general Hamiltonians $H$ and functions $a$.