ON THE VANISHING DISCOUNT PROBLEM FROM THE NEGATIVE DIRECTION

The paper explicitly shows that for the negative-discount equation (Aλ): −λu + H(x,Du) = c(H) on a closed manifold, uniqueness generally fails when λ > 0 (e.g., −u + |u'|^2/2 = 0 on T^1 admits multiple viscosity solutions), and therefore one must study the minimal solution u^-_λ among all viscosity solutions (S^-_λ) . In contrast, the candidate solution asserts uniqueness for (Aλ) via a comparison principle and concludes S^-_λ is a singleton; this contradicts the paper’s statement and example. The candidate also maps u^-_λ to v_λ = −u^-_λ and treats v_λ as the unique viscosity solution of the positive-discount problem λv + Ĥ(x,Dv) = c(H), invoking DFIZ; however, for the correspondence used in the paper, viscosity solutions of (Aλ) correspond to forward weak KAM solutions of (Bλ), not to viscosity solutions of (Bλ) . The paper’s main theorem (uniform convergence of u^-_λ to the critical solution u^-_0 that vanishes on the Aubry set under H(x,0) ≤ c(H)) is established via this weak KAM/contact approach and the Aubry set uniqueness property . Hence the paper’s argument and result are correct, while the model’s proof relies on a false uniqueness claim and a misapplied reduction.