Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly increasing in $u$, and is convex and coercive in $p$. For each parameter $λ>0$, we denote by $u^λ$ the unique viscosity solution of the H-J equation $H( x,Du(x),λu(x) )=c.$ Under quite general assumptions, we prove that $u^λ$ converges uniformly, as $λ$ tends to zero, to a specific solution of the critical H-J equation $H(x,Du(x),0)=c.$ We also characterize the limit solution in terms of Peierls barrier and Mather measures.

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