On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

The paper proves convergence of the negative-discount ground state u^-_λ to a critical solution u^-_0 under a Mañé-generic hypothesis (and, more generally, under condition (⋆)), via a variational/Lax–Oleinik framework . However, by the simple viscosity-solution transformation v_λ := −u_λ and Ĥ(x,p) := H(x,−p), the negative-discount PDE −λu + H(x,Du) = c(H) is equivalent to the proper positive-discount PDE λv + Ĥ(x,Dv) = c(H), which, for Tonelli Hamiltonians on closed manifolds, has a unique viscosity solution for each λ>0 and admits the Davini–Fathi–Iturriaga–Zavidovique vanishing-discount limit as λ→0+ (hence u_λ → u_0 uniformly) without any genericity assumption. The paper itself notes that û^-_λ := −u^+_λ is the unique viscosity solution of the positive-discount equation for Ĥ (thus u^+_λ solves the negative-discount PDE) and constructs u^-_λ as another viscosity solution via a backward operator, while stating that solutions to (HJ−_λ) “exist but [are] unnecessarily unique” . This non-uniqueness claim conflicts with the equivalence to the unique positive-discount solution; in particular, if u^-_λ and u^+_λ are both viscosity solutions of (HJ−_λ), then −u^-_λ and −u^+_λ would be two viscosity solutions of the proper positive-discount PDE, contradicting uniqueness. Consequently, the genericity assumption is not needed for convergence, and the paper’s central PDE convergence statement is strictly weaker than what follows from the standard transform plus DFIZ. The paper’s dynamical interpretation of the limit (conjugacy with u^+_0) remains a useful contribution, but the convergence result can be obtained in full generality.